Sluglings are baby slugs. The only one seen on video is a very small with red and yellow colouring.
If a Slugslinger wants to raise one, it has to raise it from this stage. It must be very hard, and that must be why it's hardly mentioned and no more than one slinger seems to own one. But if any others do, they wouldn't want to be showing it off. Not many would have enough patience and a big enough bond with their slugs to want to take care of one of them anyway (ex: Pronto, Blakk, Mr. Saturday...).