Okay, I got my books on hand now and I realized a couple of mistakes. Shoutout to the commenter on my answer for the help there. Pretty rookie mistake but that's what happens when your books aren't on hand. With that being said, I'm putting the grand total back to 6 (or 7 with loophole).
Sentinel/Beastmaster hybrid class: 2 animals
Multiclass to Artificer: Artificer gives you a bunch of abilities that will allow you to temporarily animate constructs, so kinda "animals a character can control." You could choose any arcane class, but artificer is best because it is a prerequisite for Familiar Bloodsmith. Also, there are artificer feats to upgrade your familiar.
Familiar Bloodsmith paragon path: 1 animal
Fey Beast Tamer background: 1 animal
Mount: 1 animal
Elemental Companion feat (prerequisite: Born of the Elements feat): 1 animal
Arcane Familiar feat: 1 animal. (This requires exploiting a sort of loophole your DM might allow: the prerequisite for Elemental Companion says you cannot have Arcane Familiar, but Arcane Familiar does not have a prerequisite restricting Elemental Companion. This is due to the publication dates, and the restriction surely would be there had they come out at the same time, but if you go by wording alone, and if your DM allows it, you could do this.)
And there you go, number wise!
I do know of some items, all from Dragon Magazine 377 to make your familiars a bit better too, each providing small buffs or some mobility/utility: Safe Retreat, Charm of Protection, Lucky Charm, Familiar's Cowl, and Homing Cowl.
The one build I have heard about based around pets is the Blade Dragoon build, but it only incorporates a few familiars.
While pets from the themes or classes are really useful in combat, familiars are based almost entirely on utility. As a result, you can't really make a solid build out of it, although it is cool for roleplaying.
Aside from full builds, you can take some familiar or companion feats to buff them up more, but again, companions and familiars are utility or scouting units.