Is it possible to enroll Apple Developer program without ID verification?

In general, it seems to be a good idea that Apple is asking for your ID when you're trying to enroll with a different name than your own (I guess this is what you meant by a different name - unless you have two names somehow).

I have never heard of Apple asking for an ID, and then just somehow agreeing to go without ID for no apparent reason. But you could ask Apple to be allowed to enroll without showing an ID - they are afterall the only ones that can determine if you can enroll without it.

If you have problems with the ID, you might consider creating a limited liability company and applying for the developer program with that. However, it is usually not possible to create such a company without identifying yourself.