Short answer seems to be that there is such a category and it includes most respiratory viruses, including coronaviruses:
Although the severe acute respiratory syndrome-associated coronavirus (SARS-CoV) epidemic was controlled by nonvaccine measures, coronaviruses remain a major threat to human health. The design of optimal coronavirus vaccines therefore remains a priority. Such vaccines present major challenges: coronavirus immunity often wanes rapidly, individuals needing to be protected include the elderly, and vaccines may exacerbate rather than prevent coronavirus lung immunopathology.
Alas this paper considers it such a well-known fact that they don't actually bother to cite any studies in support of that particular claim. (So, if you want to further challenge this, Skeptics SE awaits you...)
And regarding the closest know relatives of SARS-CoV-2, that is SARS (and MERS) the data on long-term immunity is not very encouraging either, as a March 18, Nature editorial relates:
Immunity is short-lived for the coronaviruses that cause common colds; even people who have high levels of antibodies against these viruses can still become infected, says Stanley Perlman, a coronavirologist at the University of Iowa in Iowa City.
The evidence is more equivocal for the two other coronaviruses that have triggered epidemics: those that cause severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS). Perlman says his team has found that after people recover from MERS, their antibodies against the virus drop precipitously. He also says that his team has gathered data — not yet published — showing that SARS antibodies are still present in the body 15 years after infection. But it’s not clear whether this immune response is enough to prevent reinfection. “We don’t have good evidence of long-lasting immunity, but we also don’t have really good data from both SARS and MERS,” Perlman adds.
More tangential info below:
The immune system can be damaged by some pathogens making them vulnerable to others, even if you had prior exposure to those. Besides the well-known HIV which causes general immunodeficiency, measles can do this to some extent as well, and you do get cured from measles proper. This from a NYT article, I haven't checked the actual science publications:
Measles is far more dangerous than most people realize, new research shows.
The disease itself can cause a severe and sometimes deadly illness, but two new studies published on Thursday found that even when patients recover, the virus can inflict lasting harm on their immune systems.
The weakened immunity leaves a child vulnerable for several years to other dangerous infections like flu and pneumonia. The damage occurs because the virus kills cells that make antibodies, which are crucial to fighting off infections.
Scientists call the effect “immune amnesia.” During childhood, as colds, flu, stomach bugs and other illnesses come and go, the immune system forms something akin to a memory that it uses to attack those germs if they try to invade again. The measles virus erases that memory, leaving the patient prone to catching the diseases all over again.
Technically this a bit besides what you're asking about. The technical term is "immunomodulation".
Also, the adaptive response takes a bit longer than people might suspect:
These individuals were likely resusceptible to reinfection with the same strain of influenza virus due to a confluence of unusual events. First, all three were reinfected within three weeks, before their primary adaptive response had sufficiently matured. Another contributing factor was the high level of circulation of the pandemic strain. [...]
Could reinfection also occur after immunization with influenza vaccine? Yes, if the immunized individual encounters the virus before the primary antibody response matures, which occurs in 3-4 weeks. This is more likely to occur during pandemic influenza when circulation of the virus is more extensive than in non-pandemic years.
- Perez CM, Ferres M, & Labarca JA (2010). Pandemic (H1N1) 2009 Reinfection, Chile. Emerging infectious diseases, 16 (1), 156-7 PMID: 20031070
And it's suspected that it's not actually true (as claimed in the question) that prior exposure to a family of viruses, e.g. influenza, always makes subsequent infections with different influenza strains milder. In fact, there's some evidence that the W shaped mortality curve (by age) for the 1918 influenza was due to the exposure in infancy of some age groups (but not others) to a somewhat similar viruses that primed a overreaction of the immune response, which was often responsible for the rapid swelling of the lungs:
The 1918 pandemic was caused by an H1N1 virus (with a Group 1 HA). Those whose first exposure had been to a putative H3N8 (Group 2) virus that emerged in 1889 were at high risk of death . Crucially, because children born several years prior to a newly emerged IAV strain can experience that virus as their first (or among their first) IAV infections, one should not expect a clean demarcation of increased risk to coincide with the year of emergence of that H3N8 strain. Although the young-adult mortality rate from the 1918 virus has a sharp peak in those born very near 1889, it stretches back to include those born up to a decade or so prior to 1889 (Fig. 3b). Interestingly, it also stretches forward only a decade or so, possibly because a new H1 virus emerged in the early years of the 20th century, displacing the 1889 H3 virus [20, 24]. This idea is supported by the lack of evidence of anti-H3 antibodies in those born after the turn of the century—despite clear evidence of N8 reactivities until shortly before 1918—as well as by the low mortality during the 1968 H3N2 pandemic in those born before, but not after, about 1900 .
[...] Nevertheless, results indicating that the 1918 virus is unusually lethal in mice , along with the fact that introduction of all new internal proteins in the 1918 virus could have played some role in its unusual virulence (due, e.g., to absent cellular immunity to new T cell epitopes) , suggest that the overall virulence of the 1918 virus may have been affected by factors other than antigenic imprinting in childhood. These uncertainties make the recovery of archival viral strains from prior to 1918 particularly attractive: exposing experimental animals such as ferrets, pigs or mice to reconstructed versions of putative H3N8 and H1N8 viruses that may have provided distinct imprinting of different cohorts in 1918  may be the only way to resolve these questions and answer, finally, why this pandemic was so catastrophic.
Unfortunately no genomes of pre-1918 influenzas have been reconstructed insofar, so it's presently impossible to experimentally verify this kind of "kindling" hypothesis, at least for 1918 pandemic.
And more generally, it's suspected that nobody becomes "fully immune" to coronaviruses and some other respiratory viruses like the RSV:
The toll of a seasonal-flu-like coronavirus also depends on immunity — which is also scientifically uncertain. Exposure to the four endemic coronaviruses produces immunity that lasts longer than that to influenza, [Richard] Webby, an influenza expert at St. Jude Children’s Research Hospital] said, but not permanent immunity. Like respiratory syncytial virus, which can re-infect adults who had it in childhood, coronavirus immunity wanes.
“Everyone, by the time they reach adulthood, should have some immunity to some coronavirus,” said Tim Sheahan, a coronavirus researcher at University of North Carolina’s Gillings School of Global Public Health. But because it doesn’t last, older people can get reinfected. The elderly also have a higher death rate from coronaviruses such as SARS and MERS, a pattern 2019-nCoV is following.
“There is some evidence that people can be reinfected with the four coronaviruses and that there is no long-lasting immunity,” Dr. Susan Kline, an infectious disease specialist at of the University of Minnesota. “Like rhinoviruses [which cause the common cold], you could be infected multiple times over your life. You can mount an antibody response, but it wanes, so on subsequent exposure you don’t have protection.” Subsequent infections often produce milder illness, however.
I'll try to find some actual studies on this as it's the most central issue relating to the question. There are lot of papers in pubmed about "waning immunity" but most are about vaccines, so that may or may not fully answer your question. But to pick an example (of such a paper on vaccines waning):
For mumps, protection appears to wane over decades, prompting the use of additional doses of vaccine for outbreak control . Transmission models have been used to study long-term age-specific time trends of pertussis and mumps. The best-fitting models require waning immunity, leading to the inference that vaccine protection does indeed wane, as well as providing an estimate of the rate of waning [19, 20].
I think I found some research that might satisfy you a bit more, but it's alas not about influenzas etc.:
Determining the duration of protective immunity requires quantifying the magnitude and rate of loss of antibodies to different virus and vaccine antigens. A key complication is heterogeneity in both the magnitude and decay rate of responses of different individuals to a given vaccine, as well as of a given individual to different vaccines. We analyzed longitudinal data on antibody titers in 45 individuals to characterize the extent of this heterogeneity and used models to determine how it affected the longevity of protective immunity to measles, rubella, vaccinia, tetanus, and diphtheria. Our analysis showed that the magnitude of responses in different individuals varied between 12- and 200-fold (95% coverage) depending on the antigen. Heterogeneity in the magnitude and decay rate contribute comparably to variation in the longevity of protective immunity between different individuals. We found that some individuals have, on average, slightly longer-lasting memory than others—on average, they have higher antibody levels with slower decay rates. We identified different patterns for the loss of protective levels of antibodies to different vaccine and virus antigens. Specifically, we found that for the first 25 to 50 years, virtually all individuals have protective antibody titers against diphtheria and tetanus, respectively, but about 10% of the population subsequently lose protective immunity per decade. In contrast, at the outset, not all individuals had protective titers against measles, rubella, and vaccinia. However, these antibody titers wane much more slowly, with a loss of protective immunity in only 1% to 3% of the population per decade. Our results highlight the importance of long-term longitudinal studies for estimating the duration of protective immunity and suggest both how vaccines might be improved and how boosting schedules might be reevaluated.