Generally no, not enough to be able to treat exactly for that cause. Otherwise it would be a routine part of the medical diagnostics for infection in a hospital. After all, why send away for a test if you could do it more quickly and cheaply yourself. If you had the equipment/reagents on-site and the know-how a Gram stain of pus should take you <20 min and could allow you to quickly identify Gram postive from Gram negative and potentially a possible species by morphology and cause alone (e.g. Streptococcus pyogenes or Escherichia coli).
In some cases it is possible to distinguish well enough for an empirical treatment - as you noted sometimes Pseudomonas aeruginosa infection produces a tell-tale blue/green sheen and/or a fruity smell, but this is an indicator only. Not all P aeruginosa infections cause both or even one of these characteristics. Some groups of bacteria can be suspected enough for an empirical diagnosis and treatment (e.g. "Strep throat" in a child is generally caused by a Group A Streptococcus, usually Streptococcus pyogenes; skin infections are often Stapylococcus aureus but could also sometimes be Staphylococcus epidermidis or any of quite a range of species and genera), but no response to empirical treatment would need further diagnosis with species identification/culture and drug-resistance tests, and in many cases this sort of delay in treatment can result in poorer outcomes for the patient, so it is better to take a sample and get a proper identification while treating empirically.
In general infections get treated by the body in the same manner and produce the same results - inflammation, purulent material, tenderness, etc, which are not enough to distinguish the infectious cause. Anaerobes do generally have distinctive smells, but they are often not found at high enough concentrations outside of culture to be able to distinguish them for certain. However, "anaerobes" includes a huge range of species from many different genera of bacteria, ranging from facultative to obligate anaerobe, so doing a diagnosis on the basis of "anaerobe smell" is risky without looking for other potential species that might not induce a smell or visual characteristic that you can identify.