I mean, we already do tall to Shego every so often in-universe. It's how we got her as our +1 for the Gala. I imagine that preforming the Shego action will just make it so we have an official way to contact her, along with us being able to spend actions so we can ask her for help with something
This has been confirmed by the GM as not being the same thing. What we do is currently do are the usual monthly updates about her brother, giving us something of a personal relationship with Shego. What this action does is open up formal relations between DEI and Drakktech, which allows us to work on research jointly or collaborate on an issue. In terms of research, we do have combat relevant tech from the Gala that would be good to acquire, and in terms of collaboration, well, I imagine Shego has more experience with organizing PMCs and the like (since she's lacking massive Normbot swarms), so we can probably get some input on how to how to make ours more efficient (thus helping lower the DC on the action which gates our second Martial action).
As for a collaboration vs Toffee, that would be something we pursue next turn, after we have opened formal relations. At that point, we'd have completed the Investigate Toffee action, and would have that info available.