I'd like to start by saying that I'm playing 5E, and with that, my question is whether there's has ever been in D&D in general (1E-5E) a ruleset for dealing with the idea of having Vanity Pets. I.e., non-combat capable pets. So the kind of pet one might have if they're not a Beast Master.
Is there anything stating that those are against the rules, or in any way break the game? I mean, if you would like to have it as part of your backstory, and/or for purely RPing purposes? I'm trying to think, but I can't really see if that would give any single player an edge over another.
I ask because the group I play with is preparing to start another campaign after the one we're on finishes, and the two people who are co-GMing, aren't sold on the idea of a non-combat pet, mostly coming from a rule-based perspective. As I understand, the main concern is that I might be getting something out of it that they can't see yet, something that would give me a work around the rules, or some sort of advantage. Which is way over thinking it, since I just want some sort of vanity pet that follows my character around 'cuz backstory…
That said, this is something that I am curious about for my own reasons, because this is something that I've been thinking about trying to encourage players to think about for a campaign I'm putting together. And if this does indeed cause an issue, I'd like to be able to see it so that I don't end up breaking the game for my players when I end up GMing myself.
Best Answer
There is nothing in the 5e rules (at least that I'm aware of) that either allows or prevents the acquisition of a non-combat non-magical pet.
That means that whether or not you can have a pet is completely and totally up to your DM.
As to whether or not there is a balance issue related to having a pet, I would not be very worried about it. It may occasionally provide a very small benefit (I'd maybe let it provide advantage once a day if I wanted something explicitly mechanical for it), but in general it's not going to be important and is merely a flavor element for a character.
That said, if you did want a mechanical element to a pet, the Find Familiar spell would be a good place to look to get a non-combat effective pet that does have some mechanical utility.
In summary, there's nothing in the rules that allows or prevents this, so consult your DM, but do so knowing that it's probably not going to be a mechanical concern (especially if it's spelled out explicitly that it can't be). If there is a need for more mechanical weight, then find familiar is a good model.
On the more general aspect, I can only speak to 4e, but it had a theme called Animal Master that gave you an animal companion that was basically just a vanity pet with a small mechanical benefit (you could use it to gain combat advantage once per day). It was a minion, so not very combat effective and was mostly a flavor piece with a minor benefit.