I'd recommend races like tabaxi be played like humans with a different set of habits/tics.
We tap our feet or zone out when we're bored. Elves don't sleep, they go into a trance. Meanwhile, a tabaxi could either doze off (still listening to the world around them), fixate on some moving target, or play with whatever object is near. Or get drunk way too quickly since their livers don't process alcohol very well. Simple stuff like that to give them flavor without making them too alien. They're still sentient and operating productively in a world of humans, after all.
That all goes out of the window if you're playing something like Dark Sun's races, though. If they're specifically stated to have very strong racial mental characteristics (like the dwarves being turbo-autists), then play that. Otherwise, stick to the general behavioral description but try not to be annoying.
My unpopular opinion when it comes to the DM side of things is to mostly avoid the racism angle. I get that Mutt's Law is a thing, but I find race relations is overplayed in real life and media. It's best kept as a card to play rarely and subtlety, and is best used for jokes and adventure hooks rather than an end to itself. The old "shopkeeper doubles his prices when there's an elf in the party" is dumb, easily worked around, and doesn't add anything fun to the game. Having the shop keeper go to attack the elf, and reveal an elf that robbed him recently and elves look alike to him, that's an adventure hook. The tabaxi in a mostly human village that struggles to find a hat that fits is a fun running joke and can be used to nudge that player where you want.
The racism angle gets boring when it's universal, generic and/or unchanging. Like, when everybody has the same reactions, and these reactions only really exist because they have a stat or resource-based consequence. The whole "oh, you're an elf? It's 50 gold pieces instead of 25 now. Take it or leave it" thing you mentioned. In most cases the average fantasy person's reaction to an uncommon race will be curiosity tempered by whatever rumors they've heard about these people. Lots of glances, clutching their coin bags, telling their kids to stop staring, yes. Yes, in a region that is still recovering from a raid by the local wood elf raid the party's knife-ear might get some abuse thrown his way but at least there's a clear reason for it.
On the other hand, some races should be copping more abuse or drawing more attention by default in some areas. If goblins in that world are murderous little bastards, the goblin in the party should be used to sleeping in barns, to the point it's not even a point of contention in the game. Same with the tiefling in a setting where everybody knows they're descended from people who had pacts with demons. "You get to the inn and spend the night" would have "and the Goblin Rogue and Tiefling Warlock sleep with the horses" as an unspoken assumption. After all, there's no point in roleplaying what's commonplace.
Why do that if it doesn't usually have an effect, though? Easy: it gives you opportunities to do fun things when they
do matter. Following the example: besides fun little roleplaying moments like the goblin going "holy shit, you guys sleep on meringue!" when he gets the chance to sleep on a fancy bed, you can also have situations where the people sleeping in the barn get to see things the rest of the party doesn't. Say, the inn catches on fire overnight. While the rest of the party is busy trying to put out the fire and make sure the innkeeper and his family are alive, the two who were sleeping at the barn can see the arsonists running away and either track or tail them to their hideout to complete the adventure hook. Or the party heads in to an elven apothecary and the Goliath is told to stay outside because "a clumsy oaf like that will just break my vials"... only to see some plot-relevant commotion or suspicious activity while he waits.
Really, people just need to be more creative with these things. Not that there's usually much opportunity for it: everybody talks about how they want deep and character-driven stories with plenty of social interaction, but 90% of the time they're crawling dungeons anyway. And the gelatinous cube sliding down the hallway doesn't care if you're a human or a dragonborn.
