What does the idiom "you've got me " means? Does it have the same meaning as "you've got me there"?

And is there any idioms such as" I've got your back" meaning" I'll support you"?
1. You've got me means "You have stumped me; I have no answer." You've got me there can be basically the same.

2. 'I've got your back' is interesting. It implies not only support, but protection. The literal use is often in cop movies, the figurative use often appears in business when one person is against another. 'I'll back you up' is similar, but loses the idea of protection. 'I'll watch out for you' is the closest other idiom I can think of. There's also "I'll cover you" and "I'll cover for you", which usually mean that the speaker will do some work on the other's behalf. "Take someone under your wing" is also similar. Do these help?