No, it is not possible. You either have the setting that allows users to post anonymously or you don't. That's it.
However, there is something somewhat related (actually not that much) implemented in the core that not so many people is aware of. Let's go step by step.
1. Allow anonymous posting
2. Post an anonymous question. You will see that in the question, below, it says something like this: asked 6 seconds ago by me. So even if you are not logged in you're still recognized as a specified a user. If another user (logged in or not) looks at your question the same question will display asked 6 seconds ago by anonymous.
3. Register using the same browser session (and with the cookies not cleared since the post creation)
4. Go to the question you have previously posted as an anonymous user. You will see that it now says that it is posted by an anonymous user but if you look below you will see an uncommon action with the text I wrote this.
5. Click that button. That will claim the question and assign it to your user.