I'd propose a three tiered approach to determining how to make this work:
1. Has the character encountered this particular creature pre-game?
This is where detailed back stories are great. How did they develop this favored enemy?
If through study, would this creature have been described in enough detail for the character to recognize them?
Diedre's mother was a dragon hunter. Her grandmother was a dragon hunter. Her great grandmother was a dragon hunter. She was honestly sick of it. So when, at 16, she found an abandoned warlock's hut with a small library on Mind Flayers, Beholders and their subordinates, she decided THIS would be her quarry. No more lizard bothering.
If through direct experience, did they experience this particular kind?
When Sindar was just six years old, his uncle accidentally opened a portal from the abyss into his study. His family spent the next month beating them back, and when the portal was finally closed again, his uncle (or his body) was nowhere to be found.
If the creature isn't explicitly identified in the back story, you could make a judgement on how rare the creature is and how much exposure the character had, or you could move on to:
2. Does a passive check reveal their identity?
Give the creature an identifying DC based on how rare they are, and then apply a passive check (10 + character's relevant modifier) with advantage (+5) because of Favored Enemy. Typically a History check, though you could use your discretion to determine what would be most appropriate in a specific situation.
Sindar, above, might get a perception check to see if he recognizes the visual from the encounter in his childhood, for instance.
If they fail the passive check, could they get a last shot at recognising the creature? Well...
3. Does the player ask for an active check?
If after your description the player goes "Oooh, black and slimy, are they? Do I recognise them because of *checks character sheet* Favored Enemy?" You could choose to give them a last shot. Have the character roll an active check, just like the passive one before, except instead of adding 5 for advantage, roll twice and take the higher result.
What if they just don't recognize them? Can I still give them the bonuses?
If after that they still don't know what it is, the choice of whether to secretly grant them their bonuses depends on how you interpret how Favored Enemy gives them these bonuses.
If it's the fact that a greater interest has, over time, granted greater insight into their strengths, weaknesses and tactics (which is how I'd interpret it) it doesn't really make sense to grant these benefits unknowingly. Though you may prompt the player to make an active check part-way through a battle based on the idea that they may recognize them by their strengths and/or tactics...
If it's some mystical affinity they have developed through their attunedness to quarry they prefer to hunt, then they may have those bonuses anyway, and respond intuitively to their enemy's strengths and weaknesses.