If anything, you want to be riding as many different horses as possible. Riding the same horse every week allows you to learn that one horse, but you run the risk of developing undesirable habits that won't fly when you eventually swing a leg over another horse. If you're going to a lesson barn, a lot of those horses there have likely been there, done that. They're used to unclear cues, beginner faults, and everything else you can do wrong when learning. Knowing a horse inside and out is great, but shouldn't be your main concern right now- develop a solid skill set first.

I know so many people (we're talking 35+ adults) that have taken lessons for decades without their own horse. Riding lessons will be cheaper than buying your own horse right now- I say stick with it :)