I try to name my toons something that's generally RPG-ish but are fitting for this game. I just try to make names that are relatively easy for other players to read or pronounce. I avoid jokey names because they get old and there is always someone who won't get the joke, no matter how obvious or mild or un-edgy your sense of humor may be, and it costs 2500 crowns to change a name.
I do have one toon that has an inside joke from another fandom rolled up in an ESO lore-friendly name and I made that toon 3 years ago and even though it's so ridiculously obvious to anyone from that particular fandom, I'm still waiting for another ESO player to go, "Oh, hey, is that a reference to such-and-such?" This is what I mean by people not getting joke names.
That's not cringey. People come from all backgrounds, and one's culture doesn't always match one's "race".My first character's name was kind of cringey, in that the racial connections of his first name (Khajiit) don't match his actual race (Breton). I just pretend he was adopted and named as a baby by a mixed couple-- Khajiit and Argonian, perhaps?
I used to integrate my name and a pet's name but now I use lore friendly ones from a name generator for each race. My new Breton for example is Farstian Arius, which I got from a generator. I also have an Imperial that I mixed with a Roman god's name and a generated one: Pluto Derannes. I also have a Altmer warden healer named Glaen, which I got from a generator too.