The first thing to say about paladins is, as always, that no, they should not need any connection to any god or religion.
They are absolutely better as a power born of the conviction born with an oath made, regardless of what to.
Someone could obviously make that oath to a god, but even then it still would not be that god giving them power. Just their conviction, same as any other paladin.
If you actually get your power from a god, you're a cleric. Said and done. Unless you made a deal for it, then you're a warlock. (But those don't generally go with full gods, only comparatively lesser entities.)
A particular paladin being devoted to a god, in terms of their power, is incidental. It's still the same power they'd get if devoted to anything else.
And there's nothing out of place with that conviction forming power in the world of DnD. We've already got Kuo-Toa literally creating gods via belief ffs.

Someone said something earlier about paladins being unique in their capacity to lose their power, and/or restrictions on it?
In BG3 that seems to be the case, but in DnD in general that's very much not true.
It's worth remembering that Clerics and Warlocks are both highly liable to lose their power too. Clerics are the most at risk, if any class is.
A cleric is granted use of a gods power. That only lasts as long as the god wants. It could stop on a whim, let alone after the cleric doing something the god doesn't like. Warlocks could do something that violates the pact, and have the power they were given taken away.
...if you factor in amnesia you could also suggest wizards can lose their power, since they have to actually know what they're doing. But that's pushing it.

5e, at least according to the subclass page on dndbeyond.com, makes the mistake of stating an oath-breaker should be evil. Which also at least somewhat implies any non oath-breaker paladin is good.
Which would be a ridiculous restriction on both counts.
Most of the subclasses could be formed with an evil person making an evil oath. In which case, that paladin would become an oath-breaker by becoming a better person, chances are. There should be good oath breakers, both in BG3 and 5e.
The oath-breaker npc in game seems like an example of this, from what little I saw of them. [My friend wanted to be a breaker in a multiplayer run, so I got to spectate conversations with the npc about why they broke their oath, and it sure sounds like "I stopped the evil man from doing evil things, and refused to do them myself". Making him an oath breaker by being a hero. Go figure. For my part I've only played paladin without breaking my oath so far, barring a few moments where I saved and reset just to test if an action would break it, to make sure I understood the oath text the game provides.

Focusing more on the game now:
The text on the oaths we get in that book and subclass choice is a level of understanding I can reasonably expect from a player. But from the character?
I would expect the character who's actually living that life to have a much better and deeper understanding of their oath than a player would. To be far more capable of realizing that some action or choice would violate their oath.
And since anything the character knows, the player should be told, that means that in most if not all cases, the game should simply indicate that an option would break the oath. It would be obvious to the character, so it should be obvious to the player.
Would you really expect paladins to violate their oaths accidentally? No, they'd make that choice knowing exactly what they're doing, at least most of the time.
Also it's a bit too easy to steal an item entirely by accident, not having meant to pick it up at all. A player unintentionally looting when they were just trying a movement command is hardly equatable to an oathbreaking action.

I'm not sure how to feel about using a companion to do an action that would have been oath-breaking, in order to not break the oath.
On one hand, it is technically true that they did it, not you. On the other hand, they only did it because you made them.
When you order someone to do something, that doesn't pawn all blame off on them. You're both responsible. So logically, if my Oath of "never kill anybody ever" Tav sits there while I control Astarion to assassinate somebody, it's still really Tav that killed them, so it would be reasonable for the Oath to break. Astarian only acts as an extention of Tav.
In multiplayer, of course, the character and companions controlled by other players are not to be considered under your control, not extensions of you. So they could do something without it affecting your oath. We don't need players fighting over that.

But without a doubt, the most ridiculous thing about paladins in this game is:
Respeccing out of paladin, doing the oath breaking act, then respeccing back into paladin, so that the oath never breaks.
Just no. That absolutely should not work.
An oath, especially one with the conviction necessary to power a whole class, is not something you can just make and unmake willy nilly.
Once a player (character) takes the Paladin class, they should be subject to the oath of the chosen subclass for the rest of the game.
Respeccing out of paladin should not and can not be considered freedom from the oath, just an oathbound person who's not currently actively using the associated power.
Committing an oath breaking act while respecced out of paladin should still engage the oath breaking story. If not immediately, then as soon as the player ever tries to spec back into paladin. That plotflag needs to trigger regardless of what classes are currently active.
I have no comments on the redemption process available in game, I haven't seen it.