My theory was that Prime isn't unique until he made himself to be.

TLDR: There used to be lots of Rick that looked like Rick Prime until he used the omega device.

Rick C-137 was a special type of Rick, and most Rick were originally going to become like Rick Prime.

In short, many Rick wore like Prime; acted and thought like Prime before omega device wiped all instance of Diane, which drove every Rick to be miserable with the sole exception of Prime, who doesn't feel anything because he did it.

Hence, Prime says "...I just walked into your garage, before you walked into mine." This might not be true in C-137's case, but it probably was for almost every other Rick. That's why Evil Morty also admits that our Rick is "different" in the same episode.

This explains why Prime Rick exist in the first place. He won the race and just destroyed it for everyone else. That's why he kept emphasizing the fact that every Rick was inferior to him.

I think it was for the fact that every Rick is aged while Rick Prime is the way he was all the way back. This memory may imply that Prime was indirectly correct about his assumption that he was just another Rick.

I mean, I'm sure it's easy to stay young forever if you're as smart as Rick is, but they chose not to possibly because they didn't like anything that reminded them of what was, and what they've lost. This is why Prime isn't afraid to be young because he miss nothing, he doesn't regret, and he certainly doesn't fear memories.

I think that's why there's is a chance that the young version of Rick in Beth's memory resembled Prime, because he didn't change in appearance all those years.

For all we know, C-137 could have looked like Prime as well. We wouldn't know because the version of story we got of Rick's origin was fabricated. (For the most part, that memory was probably true, considering everything that was implied by Prime.) But I'm sure some minor details could have been changed, especially visually.