Multiple familiars are possible in Pathfinder, but only if they are gained from the same class. The simplest way to gain multiple familiars is as a Cleric.
Clerics gain 2 Domains at 1st level. The Crocodile domain grants a Dwarf Caiman as a Familiar as one of its domain abilities, likewise the Monkey Domain grants a Monkey familiar as one of its domain abilities. Thus a Cleric with these two domains would start with both a dwarf caiman familiar and a monkey familiar, which would both advance according to your cleric level + levels in any other class that grants a familiar.
You can also get familiars from the Vermin Domain, Eagle Domain, Frog Domain, Plane of Air/Earth/Fire/Water Domains, and Serpent Domain -- all of which are species specific.
If you want to really stack familiars, the 3rd-party Magus archetype the "Tovenaar" grants 2 Domains at 1st level, another at 2nd, and more as you advance in levels, which could grant you all of the above, plus an additional familiar if you take the Familiar magus arcana.