Zoomart on Abashidze street 78 (located very conveniently close to the Vake park) has been our favorite pet food store (actually the only one) for several years already. We came here regularly to buy special cats dry food.
According to Zoomart website, there are already 11 stores across Tbilisi (2018).
At Zoomart you will find brands like Canvit, ferplast, Vitakraft, Brekkies, catzone, elmubas, Versele-Laga, Advance, Kong, Acana, Morando, Animology e.t.c
Together with Natia Beridze Veterinary, these are two of the pet-related places I'm familiar with in Tbilisi