Allow me to summarize my entire argument in a lovely little syllogism for ya:
1. If an integration between GOG and Discord would be largely profitable for both, both would be interested in it.
2. An integration between GOG and Discord would not be largely profitable for GOG.
A. If a large number of GOG's users would be interested in an integration between GOG and Discord, an integration would be profitable.
B. A large number of GOG's users would not be interested in an integration between GOG and Discord.
C. An integration between GOG and Discord would not be largely profitable for GOG.
3. GOG is not interested in an integration between it and Discord.

The reason for B is that, for one thing, there isn't very many multiplayer games on here which would require use of Discord or some other similar voicechat service. If you look into other threads (for example, currently, the "Virtual Items" thread), you'll see why there's not many multiplayer games here: GOG's userbase doesn't want them, considering them to be DRM of a sort.
For another thing, why is an integration even necessary? Why would you and the other person not just make a Discord account and use it separate from GOG? I use Discord and play a lot of multiplayer games with other users on here, and that's exactly what we do.

Networking. Just because we have the public forums doesn't mean organizing players is easy. We could have community servers with a public channel for users to discuss, ask questions, and generate ideas for things to do in!

That's not a bad idea, but there's still no reason to connect it with GOG. Why don't you or some other user set up a channel and make a thread about it? Shouldn't be that hard to correlate the GOG and Discord accounts.
Unless of course by "connect", you mean that the thing would be officially run by GOG themselves, in which case I think they already do something like that with Twitch.

I meant connect as in the integration feature that Discord has with Youtube Gaming, Twitch, and other services.

Didn't know it had that.
In that case, you have to ask yourself how big those services are in comparison to GOG, and there's your answer.

What specific type of integration are you looking for? If you mean something like how you can attach Discord to your Steam/Facebook/Twitch/etc. accounts to access your contact lists and automatically add them to discord, keep track of online status and that sort of thing, it's really only a matter of GOG providing a public API for developers of 3rd party applications to be able to query the GOG platform for this information and then letting 3rd party application developers actually use it. The Galaxy API may already have this kind of information available, but official use of the Galaxy API is restricted only to partners so it's hard for people like this to really know what GOG's rules or processes are for accessing it as they have been notably private about it to date.

I think that one can safely presume that Discord has a big enough company behind it and that GOG is quite aware of them and their product because many GOG employees use it and it has shown up in photos taken at GOG etc. so it isn't an "unknown, we don't know how many people use this" or anything like that. It's safe to assume GOG is well aware how widely used Discord is and what benefits people have from using it.

What we don't know is whether Discord's company has officially contacted GOG to inquire about becoming a partner and gaining access to GOG's private Galaxy APIs, and that would be what has to happen in order for any form of integration between these two services to occur.

Since GOG's APIs are both proprietary and non-public, we don't even know if they support the types of interfaces that would be necessary for Discord to have similar integration on GOG as it does with Steam/Twitch/and other platforms, but that would be a pre-requisite as well.

The bigger question about such integration is not really so much about the GOG side other than knowing whether they have the infrastructure needed on their end and whether it is available to Discord to use, but rather the bigger question is whether or not the Discord company has interest themselves in adding support to integrate with GOG, and where such support might sit on their future feature priorities list.

On the public forum side of things, we do not have the level of open engagement with GOG insiders to find out much about the GOG end of things at the moment I don't believe, however Discord itself has a public discord server where people can ask all kinds of questions, report bugs and other stuff, and I think they also have forums and a public bug report/feature request tracker, so one might contact the Discord developer community to ask them about whether they have considered adding GOG integration support similar to what is present for other gaming and social platforms, and where such a feature might sit on their priority list, and whether it is currently technically feasible (if such APIs actually exist).

That's most likely where the answer to this will lay.

...I uh, mostly maybe meant username display and maybe minor things like that.