Does the ID need to have some sort of strict formatting? If not you could create a new text field that's marked read only to store the ID. Next using CB Auto Actions acting on the after registration trigger (both frontend and backend triggers) and a Field action to set your new text fields values as follows.

AB[user_id]

Each user is already given a unique ID (user_id), but it's not searchable or customizable. This would push it to a text field so additional characters can be added and it can be searched.

Your new field would just be a standard text field so searching on it would behave the same as any other text field. First it needs to be set to display on profile then it needs to be marked searchable.

You need to ensure your userlist is set to allow searching of it. By default a userlist only allows searching of fields that are on the userlist. So you'd need to put it in one of the columns. The alternative is to adjust the search criteria so you can specifically select what fields are searchable. This is configured on the Parameters > Search tab found while editing your userlist.