Does a player wearing a Ring of Protection +1 and standing in a Paladin's Aura of Protection (bonus equal to Charisma modifier, +2 in my case) get a +3 bonus to their saving throws? Or do they only get to use the maximum protection of +2?
Yes they stack
The two features are completely independent and have no restrictions that would prevent them from stacking.
There is a rule preventing multiple bonuses from sources with the same name (see DMG p. 252 Combining Game Effects). This would be relevant if both things had the same name but they are called Aura of Protection and Ring of Protection, so similar but distinct.
Magic Items are generally not considered in the balance and progression of PCs, so they can potentially make characters leave the limits of Bounded Accuracy. But for the Ring of Protection, the +1 AC is much more likely to pose problems in this regard than +1 on Saving Throws.
Features with different names stack.
The Dungeon Master’s Guide has a rule about stacking features together like this:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.
-Dungeon Master’s Guide p. 252
Since Ring of Protection the magic item and Aura of Protection the Paladin feature are different game features, they can stack. As long as you can get yourself into circumstances where both features are active, so in your case just wearing the ring inside the aura, you can benefit from both.