The game runs fine in single player, but if I try to play multiplayer, it crashes immediately, has anyone else had this issue? I've disabled my mods one by one, but until there are none, it crashes the second I hit "Play Multiplayer".

I was under the impression that mods worked fine online.

Mods are always an issue in synchronized multiplayer, as others probably don't have these mods, changes to the game world are not visible/applied to them, nor does the server understand, where specific data comes from. Depending on the mods you had, they might have changed values inside your save game which cannot be evaluated/certified by the server, sending rubbish back as a result and crashing the client in the process.
You might be able to cleanse your save file by disable all mods and than kind of visit any system that potentially have mod-alterations in it. By visiting without mods, the alterations 'might' be set to 'normal' values again. But that's just a shot into the blue from my side. I never had this issue (I also only have 'quite please' active which doesn't alter any content.
As always, before trying, make a backup of your save file.

I think this is more about the old problem that (on GOG) it is not (any longer) possible to select a 'multiplayer game' from the lobby. Or rather, one would need more information here about what exactly doesn't work (does the game also crash when the network options are/will be activated in the game? What's going on in the anomaly - are there no other players?)

Because actually there is no pure single-player mode - unless you turn off the corresponding settings in the network options. The option 'Play Game' also leads to a multiplayer game if you don't deactivate the network settings.

The selection in the lobby actually only determines the starting point (in the system of the person you select). Joining friends is currently easier by appointment in the anomaly or at another point.

well, I never connected to the internet with NMS. Firewall rules are a blessing against rogue applications who doesn't value privacy. So, there is an 'enforced' single player mode :)