i've been trying to play the the multiplayer which i assume is done through the gog galaxy overlay to invite friends but when i shift tab it does not appear but my mouse appears as though it is open. my gog galaxy version 1.2.57.
the overlay is working on system shock 2 launched through gog galaxy

Yep, same here. I was wondering how I'd invite a friend of mine to play with me, and overlay doesn't work, nor do we find each other in the Anomaly, despite others being around... and the invite button in the network settings doesn't do anything, either.

How do I play with my actual friend...? Making new friends is fine and all, but what for if I won't be able to play with specifically them the next time, either...?

This is something that is associated with the changes made for Vulkan months ago. GOG support staff tried to help me for a couple of weeks and finally had to throw in the towel as this appears to be a programming issue as NMS has these keys assigned to other game controls.

See https://www.gog.com/forum/no_mans_sky/galaxy_overlay_after_vulkan_177_vulkan

Here for the direct reporting of bugs:
https://hellogames.zendesk.com/hc/en-us/requests/new?ticket_form_id=167605

I rather blame GoG, cause Valve and EA managed to get their overlays working in Vulkan games, regardless if those games have the same bindings as the overlay (being Valve the best example with their own games).

Adding to that, the Steam Overlay is working fine on NMS Steam Release, which proves my point further.

If this was a problem with the game and Vulkan itself, it would impact Steam as well, GoG that needs to work on their overlay to fix compatibility (they can't expect the developers to waste time and resources on a feature they not responsible for).

I mean if you look closely, the GoG GALAXY overlay doesn't work in the majorly of games, even with CD PROJEKT own games there's problems (The Witcher 3).

I personally never saw the GoG GALAXY overlay once, and yes I have it enabled and it never worked in any game (as I said before, not even The Witcher 3).