The genetics of both games are fixed, so once you learn a specific combination, you'll be able to apply that knowledge to any future breeding, no matter how many times you play or start over.
Hrm. That's...faintly disappointing, I must confess. Part of the appeal for me is investigating which plant combos create which new plants. I made up a little spreadsheet and everything. Kind of a bummer to know that once I get it all filled out, I'm done, and future games will simply be me consulting my spreadsheet.
On the other hand, I may feel differently after filling out the spreadsheet with 500+ fish!

Are the plants and/or fish which are the Magic Plants/Fish randomized with a new game? Or will the same species of Plant/Fish always be Magic in every game?