As you say, you can't apply for your own enhanced DBS check. If you have one, you can apply to join the DBS Update Service within 30 days of receiving it. Obviously, that transfers the cost from them to you. If you have a recent check, that also takes the cost away from them, which is probably what they really want. If you just don't have one, though, then it's up to them to decide whether or not to pay up for the check.

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If you are to punish a man retributively you must injure him. If you are to reform him you must improve him. And men are not improved by injuries. (George Bernard Shaw)

I am increasingly seeing job adverts requesting to show or produce a recent enhanced DBS. My question is not about the legitimacy of this request, but whether they can ask for something that the end user cannot legally have.

As long as I know, enhanced DBS's can only be requested by the employer and they then stay with the employer.

How can then an employer ask for an enhanced DBS if: a) one has never been asked one, or b) they have, but it is with the current/former employer?