Is it possible to step through TSQL that uses variables, in SSMS?
Every time you submit text from SSMS the batch will end when that finishes executing and any variables will go out of scope. 

The only way to do this is using the debugger as illustrated below. Then you can step through line by line using the F10 or F11 keys. This should only be used in a development environment.

Also (as pointed out by @clifton_h) this feature has been removed from SSMS 18.x but is still available in the latest GA build (currently 17.9.1)