In D&D 5e, I can't find a skill check for identifying monsters. (Other than Arcana for planar monsters). Can PCs identify monsters, and if so, what skill should they use? (Or is it just a Wis/Int check?)
Sometimes the Rules Are Guidelines...
According to the Player's Handbook, "[Y]ou can use [Knowledge skills] to identify monsters and their special powers or vulnerabilities. In general, the DC of such a check equals 10 + the monster's HD. A successful check allows you to remember a bit of useful information about that monster" (78). And there just ain't further guidelines; I mean, there's that ominously pregnant in general squeezed in there to allow DM fudging, of course, but otherwise the DM's on his own.
...And Then There's Lore
That changes in, I think (and comments can totally correct me if I'm wrong), July 2006, when Monster Manual IV included a lore entry for each creature. In the Manual's introduction under the Lore heading, it says of the Player's Handbook's use of Knowledge skills to identify creatures
[t]hat [the creature identification rule] addresses specific creatures very well, but there’s more to be said about creatures of general types.... As a general rule of thumb, a DC 15 check or higher will reveal all of the base creature’s type and subtype traits as defined in the glossary. This often includes information about energy resistance or various immunities. For instance, a DC 15 Knowledge (arcana) check reveals that dragons have high hit points (12-sided HD), all good saves, and have darkvision out to 60 feet and low-light vision. They are immune to magic sleep effects and paralysis effects. They eat, sleep, and breathe.
Information specific to the creature, such as its type of damage resistance, spell-like abilities, or immunities come with the high DC check result. (6)
Then, without further preamble or explanation, each of these lore entries instead of being based on the creature's HD is based on the creature's Challenge Rating (a system later adopted by Pathfinder), with the DC starting at 10+CR, a change which makes sense to a degree yet leaves unaddressed corner cases like advanced and templated creatures.
I did some research and there was no Design & Development column (yes, they're really called that--O, you wacky Wizards!) about the Monster Manual IV, and Dave Noonan doesn't mention the lore entries in his articles here and here about the design and development of the Monster Manual V, so I don't know the rationale behind the change from identification based on HD to identification based on CR. (By the way, these are the last 2 Design & Development articles published, which is a shame.) (As a further aside, I was disappointed at the removal of both the Environment and Treasure lines in the later creature entries, but that's another issue.)
The gnoll entry in Monster Manual IV (67-71) presents gnolls of CRs of 3, 4, and 6 yet provides lore results based mathematically on the CR 1 gnoll. The design intent seems to be that when an advanced or classed creature is encountered the CR of the base creature is used to gather its lore. Templates, appear--based on the templates Lolth Touched (MM4 94), God-blooded (MM5 66), and Phantom (MM5 130)--to require a Knowledge skill check with either a DC of 15 + the creature's new CR or a DC of 15 straight up to identify the template. That's just for the template, though; identifying the underlying creature is a separate Knowledge skill check.
"But What About Dragons?"
Drow of the Underdark gives us the deep dragon (114-7), the first dragon I could find post-Monster Manual IV that has age categories and a lore entry. The Knowledge check DC for the deep dragon is 15 + the dragon's CR, implying that dragons of different CRs are different creatures therefore requiring separate Knowledge checks.
I use this, with DC = 10 + the spell's level
And I have used it since before I read the rules in XGtE with absolutely no problems at all and my group really appreciated it.
This is also exactly what passive checks are for. If you notice something your brain immediately makes the link to your knowledge and you just know, you don't need to stop and concentrate to know something, you either know it, or you don't.
This makes Counterspell more powerful than it seems to be intended in XGtE, but I think that rule sucks and having a spell countered, or countering a spell is more fun than wasting a counterspell. But then as a GM I am on the players side and like them being informed about anything and everything.
If using DC10 as the base drop the advantage rule, because that boils down to +5 and means you start on a base of 15 passively so pretty much recognise everything.
I did try DC = 15 + the spells level, and this has the effect of making non-arcane spells more difficult to counter. I didn't like this personally, but it can be a flavourful touch. The DC15 is essentially taking into account that you have advantage on your own type of spells. I think here it is a group preference thing.
I find that more information in the hands of the PCs is better than less information, and any clever use of a passive skill such as this is very much within my style of DMing, but each to their own and this is steering away from RAW.