Is Multiplying Public Keys Valid?

A quick question about Bitcoin stuff. If I do Public key * (private key), I get a public key. So, what about (same public key) * (same public key)? Is that even a thing?

Nah, you can't really do that. In ECDSA, multiplying two points isn't valid. You can only multiply a point by a scalar, which means adding that point to itself a number of times. If you try to convert that second point to a scalar, you're not actually doing what you think. It's different.

But why is point-to-point multiplication not allowed in ECDSA?

Using a public key to make a scalar isn't legit in elliptic curves. You stick to multiplying by scalars or just adding public keys together. That's how it works.

It's not just that it's undefined, it's more like there's no fast way to do it. If you had a quick method for solving DLP, sure, you'd get direct answers for point multiplication. But since figuring that out is tough, you can't easily compute products of points without running into problems.