Well, first it would help to decide whether we’re talking about mounts or animal companions, since they’re very different beasts. Most of the discussion so far is around companions though so we’ll start there.
If you want the companion to be an option early and scale into late with the same animal, you’re really going to have to create a system like 4e did where you have generic ‘bear’ template that progresses as you do. Short of swapping out animals as you level Pokémon-style it’s never going to be balanced otherwise, as others have pointed out.
If you’re okay with it being imbalanced, then this is one of those situations where I’d say the GM needs to make ramifications of the choice felt. How do they get this creature? How much to it cost to maintain? What areas are off-limits/inaccessible to the creature? How do civilized folk react to it? Your war elephant isn’t going to fit in the halls of that fortress, so it becomes a question of being situationally overpowered.
Additionally, unless the player themselves has an easy way to revive/recruit a new animal, then similar to the old ranger, the creature will be an additional resource drain on the party, particularly in regards to healing spells since they don’t have HD to spend like a PC does. This only gets worse as time goes on, since while the hypothetical bear ally can hit high CR targets, it’s really going to have issues against breath weapons and save effects.
For my money, I’d make it a feat and homebrew up some basic templates to try and keep power reasonable early and make it not obsolete late. It might also give the player some customization options for the beast as well.
As far as mounts go, it’s mostly a question of mobility, which can alternating between being a non-factor or more disruptive than the higher DPR. All of the above in-world ramifications apply, and ultimately aside from being given a mount that can perpetually fly, it’s the more manageable of the two, and already has the mounted combat feat tailored for it. Unless you’re giving your mounts attacks, then whether it’s a horse or bear is of little consequence.