Help with understanding what went wrong

In the spirit of doing the easiest fix first you could experiment with a coat of darker stain, gel stain or whatever. The problem is that you are starting with inconsistent starting points - two slightly different colors - so I'd expect them to be different only darker. Maybe it would be close enough.

I'm a big fan of seeing the wood but another easy way to get a consistent finish would be to paint it. (I cringe even as I suggest it).

I know it's a lot of work and I hate to say it but when I've come out with finishing results I didn't like I've sanded it down to bare wood and started over.
Try a different product the second time.

Congratualations. You've just discovered that birch is really hard to stain well. It is noted (notorious) for being blotchy. Welcome to the club.

As David D has answered, if you want a stained wood surface you're going to have to sand through the stain and get rid of it.

There are a couple of approaches to staining blotchy wood. One is to coat the wood with a sealer like shellac and then stain. Another is to to use something like a gel stain. You can see both here https://www.youtube.com/watch?v=XL_58_22SEA.

Note that MinWax, who you'd think know something about stains, says,

"Less expensive than hard maple, birch is often substituted for maple in furniture and kitchen cabinets. But, like hard maple, birch wood does not absorb stain evenly and should not be stained with dark colored stains. When staining, first apply a pre-stain wood conditioner, then select stains lighter in color."

There is lots of info about staining birch on the internet, so just search on "birch stain blotch" or something like it.

Also, file away the fact that lots of woods have more or less blotching issues, including maple and cherry. Do some research (or at least test panels) before staining a new wood.