Currently [[Special:Preferences]] shows a user his or her internal user ID. I'm not sure it makes sense to expose this in the user interface. It's an internal implementation detail. Plus on wikis with CentralAuth/unified login, there's additional complexity (i.e., each user has a global user ID and a local user ID on a particular wiki).
The API should expose and allow the use of a user's ID and there are separate bugs about that. This bug report is focused purely on the user interface.
Looking at how other software/sites handle this may make sense. For example, I don't think Facebook or Twitter or LinkedIn or Google Apps readily expose user IDs (apart from URLs, though those also have trended away from exposing user IDs).
This came up again in the context of https://gerrit.wikimedia.org/r/101221. I'm filing this as an unconfirmed bug. Any input welcome.