Lady: "There are other things too that I feel should be limited to a specific class or classes such as picking locks."
Ranxerox: "Is that even a thing in tabletop 5e?"
Lady: "Yes, but I guess it depends on your DM if they want to be explicit or they homebrew it."
I'm not putting words in your mouth, Lady, that is what you said, and you are incorrect.
You say it's in the player's handbook, well, cite me a source. Show me a quote.
Anyone, literally anyone, regardless of their race or class or anything else about them, can begin the game with proficiency in thieves' tools and a set of them on their person. This is not homebrew, it's right there in the PHB in plain text. Shall I cite the source for you?
I believe the perception comes from these exerpts:
![[Linked Image from i.imgur.com]](https://i.imgur.com/1xnjOkN.jpeg)
Afaik, you don't choose your proficiencies in raw in that way. The class you choose dictates some proficiencies, per quote:
Your class determines your weapon proficiencies, your saving throw proficiencies, and some of your skill and tool proficiencies.
However the background you choose etc can let the player choose additional proficiencies, which I suppose may include an alternative that gives proficiency in Thieves Tools. (It's been a while, so pardon me for errors, correct me if there are.)
Purely on the basis of classes, only Rogues have Thieves Tools assigned as a proficiency as baseline.
Edit: There's also this on page 125:
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
So if a player chooses Rogue as their class, and a background that also includes Thieves Tools, the Rogue gets to choose another proficiency instead.