Maybe, but would it be something useful for combat?
Maybe, depends on if you character would know about it. Depends on how well you roll.
And what does "trained in the respective area" mean? Maybe clerics might know some things about undead, but giants? Who fights giants and lives to tell about it? Who fights dragons and lives to tell about it? Who fights bulettes or purple worms or death knights or vampires or invisible stalkers?
I could come up with a list of possibilities. Suffice to say: there are a lot of ways to gain this information.
And how does this knowledge get passed down? Word of mouth, or does a PC have to go to adventuring college to learn about this stuff? None of the backgrounds in the PHB mention adventuring college.
Acolytes, spending long hours studying in an order.
Nobles, having access to high education.
Folk Heroes, presumably having traveled far and wide.
Entertainers, same as folk heroes, seeing out song and stories to well, entertain.
Hermits, having lived in a remote and isolated location, the natural beasts even some extreme ones may have come to accept their presence there.
Outlander, much like above, this person has wandered the world and learned about it.
Sage, who the background even states spends a lot of time studying.
Criminial, eavesdropping on conversations, stealing valuable tomes of knowledge, etc...
Honestly I feel like you're either A: not familiar with the material or B: have no imagination. I'm starting to feel like I'm being trolled.
I would think that a lot of "monster knowledge" would be distorted by the passage of time, fire side exaggerated tales, etc. It really does not make sense that PCs would be experts on monsters until they actually face them (and maybe even not then if the PCs stun lock them or some such).
Who said experts? I haven't said experts. I said "you might know something" about them. You might know that a troll's favorite snack is in fact, sauteed gnome, fat lot of good that will do you in combat though.