Edit: I'm new at this. In response to recommendations I received I've attempted to insert web citations. LMK if I did it wrong...
Yes, your immune system can recognize and protect against viruses you've been infected with in the past. That's why folks typically only get the chickenpox once. But getting a cold does NOT protect you against future colds, largely for the same reasons that a vaccine is highly unlikely (read on...).
There are multiple barriers to creating a successful vaccine against the common cold. First, as you noted, what we call 'the common cold' is really a constellation of symptoms that can be caused by over 200 different organisms 1, most of them viruses (including rhinovirus, respiratory syncytial virus, coronavirus, adenovirus) but also some bacteria (haemophilus influenzae, mycoplasma pneumoniae, and a couple others named ___ pneumoniae)2. A successful vaccine would need to protect against at least the most common ones (similar to the yearly flu vaccine which typically covers 3 strains of influenza) - a more difficult task than creating a vaccine for, say, polio (which only has 3 strains 3). Second, the reason you have to get the flu shot every year is because the virus mutates. The same goes for cold viruses - they mutate, and rather frequently 1, meaning a new vaccine would need to be developed at least annually.
Additionally colds frankly just aren't a very high priority. While it is true that young babies, the very elderly, and those with certain health conditions can suffer serious complications from a cold, most of us just get an annoying cough & nasal symptoms for a few days. Not a lot of investment is going to be made in preventing such a disease 4 (especially when there's so much money to be made in mitigating treatments).
- a pediatrician [all of the above is factual information except the $ made from cold treatments which represents my opinion ;) ]