Zack, I don't have any experience in this area as a parent, but as a child...
I grew up in a house full of animals and we were always pushing boundaries - bringing home stray cats, taking hamsters from friends who couldn't keep them, showing up with a mystery ferret we'd been keeping in the closet (yeah, that was me), "accidentally" letting the male and female hamster spend some time together... Actually, my mom might have been the biggest culprit, it was my dad who wasn't interested in being the local zoo.
I think what drove him nuts is that, no matter how much we had, there was always something else we wanted. My mom wanted a hedgehog, and a parrot, we wanted to keep ALL of the kittens the latest stray cat had birthed, not just one, and so on.
If fish are your thing and that's what you want the most, focus on those. Don't ask for chickens, hope for (more) baby goats, etc. Pick a couple of fish that you want to focus on and explain to her why they're important to you and stick with those fish.
Spend the foreseeable future focusing on what you DO have, rather than what you'd like to acquire.
Also, I agree that it's possible there is something going on that your mom feels is none of your business, but also causes her to feel some fish tanks need to go. Remember, even if you're taking care of your fish, they are costly in terms of electricity and water bills.
It's also possible she'd like to see you start saving some money, instead of spending it all on fish.
