This sounds like HomeLink and is indeed more complex. My understanding of it is that they partner with lots of companies to support their rolling/fixed codes and remotes so that they can be paired to your garage door.

I linked this in a sub comment, but the largest garage door maker in the US is Chamberlain [0] (which owns a ton of other brands) and uses known rolling code algorithms that can be decoded. [1]

[0] https://www.chamberlain.com/ [1] https://github.com/argilo/secplus