What do you have "Load" set to? By default it's set to "By Asset Only" and if it is then asset "profile.17" isn't specific enough as every profile activity post the user makes will use that asset. You need a more specific asset like "profile.17.birthday" or set "Load" to "None". Once adjusted retry then check within backend CB Activity > Activity after triggering your action to ensure the action is working.

Also keep in mind changing the activity owner will change the substitution user so [name] will likely be the name of user 17. I suggest just using an @mention usage. If you've latest CB Auto Actions installed you can use [action_user] to get the original action user id so you could do @[action_user] or [cb:userdata field="name" user="[action_user]" /].

If I enter profile.17.birthday, it congrats user 17 on his birthday, which is the wrong user.

You set Owner to 17 so it's working exactly as you've configured it.

So I've removed the owner, so it is the current user, that makes the right name appear, but then the activity is posted to the users stream, and not user 17's stream, which would be the right place, as it is user 17 posting the congrats for the user?

The asset and the owner of the activity entry are 2 separate things. For example Owner of 18 and Asset of profile.17 means the user 18 posted on the profile of 17.

If I enter nothing in the Asset, nothing is posted.
If I enter profile.[USER_ID].birthday, nothing is posted.
If I enter profile.USER_ID.birthday, things are posted, but the activity shows exactly that, and it's not substituted.

Asset is mandatory. Without it CB Activity has no idea where to display the activity.

This is very confusing, and getting very complex compared to previously versions.

Suggestions welcomed. The parameters that were available before are still available now. Type, Subtype, Parent, and Item are all just a part of Asset now. It would structure the asset as type.parent.subtype.item. Beyond that it still functions exactly the same. The parameter descriptions have also be significantly improved with asset examples even provided directly in the description.

So am I to understand that the 'asset' is a field to make the post identifiable, to avoid double posts? Or does it have a deeper meaning?

Asset is a descriptor. It describes the source. In your case profile.17 is describing the source of the activity being profile of user 17. This means you can use a more specific asset for a more specific descriptor like profile.17.birthday describes it's the birthday of profile of user 17. This also determines what stream the activity will display on.

So the remaining problem is that activities are posted to the users and not to user 17's stream to have the right name appear.
How do I accomplish to have the activity on user 17's stream, but with the users name?

Set asset to profile.17 and leave Owner as empty. This however is a profile activity. This means it will show on profile.17 AND the Owners profile. If you don't want that behavior you should use a more customized asset like "birthdays" then have a steam that outputs the "birthdays" asset.