You start the game with two tanks, Tank 1 and the selling tank. Both of those tanks can be used for breeding, but you have to be careful that you don't accidentally sell fish that are in the selling tank. You can purchase one additional tank, for a total of 3 tanks. There are no other tanks available in the game.

All versions of Fish Tycoon provide limited 'genealogy' information for fish that you have swimming in your tanks, but they do not provide a function for the kind of tracking that would enable you to organize your breeding efforts. That's why a lot of people have created spreadsheets and databases.

The Spoilers thread of the Fish Tycoon Guide section (at the bottom of the forums) contains a spreadsheet layout that's ready to use for such purposes, as well as a completed spreadsheet and a link to an online Fish Tycoon breeding database.