Yes, other IDEs (e.g., Eclipse) have user interface for underlying, typically third-party, debuggers (e.g., GDB). This is probably not implemented exactly the same way it's implemented in the CS50 IDE (or Cloud9), but the general idea is probably the same.
That is, as the user interacts with the UI controls, provided by the IDE, the IDE communicates with a debugger underneath the hood through an API (e.g., issuing commands, and parsing the output, rendering it for you in a more familiar way.)
Read more on how debuggers are (or can be) implemented in the CS50 IDE (Cloud9!)