I've got the other way - I have a player with a chromatic Dragonborn who explicitly feared - sort of an anti-folk-hero - in his back story. And a drow that the player wants to experience "what that means".
In this case if I gloss over it I'm stealing player agency and nullifying part of what they want about their characters.
Oh, totally. Obviously if this is what the player wants to deal with, then, by all means, make it an issue.
Like I said, go with what the player and the table wants and everyone's golden. Most of the time it seems to be a much bigger issue with DM's than with players. The other players often don't really care, and, after the fifteenth time they've had to deal with the ranger's pet tiger, it can get very, very stale, particularly for a group that really didn't care in the first place.
I've mentioned it before, but, a mechanic I saw in another game is the perfect solution, in my mind. It's called Backgrounding. A player gets 1-3 elements, with the permission of the GM/DM, of his or her character that get put in the Background. What that means is that while the element is there, it fades into the background and isn't a major issue. It might get mentioned in passing, but, it is never placed front and center unless the player chooses to make it so.
So, a pet simply fades to the background - either it stays outside of town or just isn't commented on, or whatever. It just becomes not a problem. Now, if the player starts using the pet in some way in game, then it's fair game. No worries. But, otherwise? The player has told the DM, "Hey, I want this because I think it's cool but, I'd rather not waste a bunch of table time on it. Can we just take it as read?"
In my Dragonheist campaign with the skeleton PC, we simply don't make it an issue. NPC's assume that the skeleton is under the control of one of the other PC's, or it's just a golem, or whatever. Again, in Waterdeep, where you have zero gravity dance halls, ghost dagger attacks, and giant golems that walk around the city, a single skeleton doesn't seem like too much of an issue.