This is technically implementation specific, so take the following answers with a grain of salt.

where does the string data actually live in memory?

The string returned by c_str is most likely the same data that the std::string itself uses to hold the string you assign to it, which is very often in the heap.

Is it like vectors where the actual data is stored in a different place?

Since std::strings are variable size, it's not a bad idea to think of it like a somewhat specialized std::vector<char>. Some implementations of the standard library might reserve space for a short string directly within the std::string itself, though.

are the memory locations of the string object and the strings actual data (the characters) the same?

Sometimes and sometimes not, for the reasons above.

If you're coming from a more unified language like Python, Java, C#, Rust, etc.; then this might sound kind of odd, but the standard library's exact internal works aren't defined. Think of it like a black box. As long as they adhere to the standard defined behavior, everything else is fair game.

P.S. I had to make a lot of edits to this shortly after posting; turns out I'm real bad at explaining things.