Sorry for the late response but I have finally been able to test it sufficiently.

What I am seeing is that each step of a multi-step job does indeed reuse the same session id. That said, the context does not transfer from one step to the next. In addition, the context does not register in sys.dm_exec_sessions which I expect it should. This may be part of the problem with transference to the subsequent steps. This could possibly boil down to the context being purged from memory as a new step starts despite the session not being closed.

To vet this out, I logged all of those details into a log table with context key, context value, session id etc.