What do you mean slots? Is it just because connections are coming from one IP and there is a limitation on their end?

Most IRC networks have a per-IP connection limit.

The slots in this case are the amount of connections that can come from the demo server (20 at the moment).

If you ran this software on your own server, you wouldn't run into the problem unless you tried to open 20 simultaneous connections to 1 IRC network.

Apparently there's a protocol that addresses this issue (WEBIRC): https://github.com/thedjpetersen/subway/issues/117

Is this actually implemented server-side on any networks?

KiwiIRC uses this protocol apparently. I asked the author about it and they said they have worked out special arrangements with some networks to implement it, and accept connections exclusively from Kiwi, I presume? After all, a password is required according to this spec: https://kiwiirc.com/docs/webirc