How do I increase my tactics slots or stacks or... whatever you call it, HOW?

You can't do that. Every pet has 4 maneuvers, every tactic battle has 4 turns, every squad has 4 pets. Your task is to build the more effective combo with the same conditions.

but, i saw other's pets while batteling. they had more slots, more than four

or..... every posible tactics are shown, right?

Pet profile lists all available maneuvers, not the chosen ones. You can see the chosen maneuvers only if you have premium subscription, and the squad is on the battlefield.