Yes projections need to be specified... The datum and ellipsoid are part of the projection. No, you won't be able to pass your custom datum along to an external service and have it recognize it automatically. That doesn't change the fact that you can define it locally and have your local tools recognize it. proj4 et al will happily let you define custom ellipsoids and register a new datum and work with projections on that ellipsoid and datum. You can view that just fine in QGIS et al.

Also, WMS is pretty niche and not too widely used partly because it supports things like this use case and is therefore quite complex. I wouldn't bring WMS into this. However, if you want to, the projection specifications it uses _also allow for custom ellipsoids_. You can happily request things in a projection with a non Earth ellipsoid -- it's just a matter of whether the WMS server supports it.