Why are girls embarrased in a bra and underwear yet eager to be seen in an even more revealing bikini?

Consent. If she's on her bikini and she's outside that means she wants to show off her body and wants to be seen. If you catch a girl in her bra and underwear and she's embarrassed then she did not want to be seen. Which also means you violated her personal space at the time she did not want to be seen.

Context is important. People usually only see your lingerie in very intimate settings, so our mind associates that clothing with that sort of vulnerability. I don't know if that makes sense.
And I've never met anyone eager to be seen in a bikini...

Bikinis arnt see thru when wet?

Well it all depends on context. If they're walking around with their pants sagging then they won't be embarrassed cause like a swimsuit they want it to be seen. If not they don't so they will be

Often times women's underwear is sheer so even though it covers just as much, if not more, than a swimsuit, you can see through the fabric.

No idea.