I thought you mean the campaign tags, my bad.
If you take a look at that url in the email, you will see it actually points to lists/subscribe-confirm url and in case you have confirmed your account earlier, you will be redirected to the update profile page like you do, which is fine.

So, to recap, first time you are redirected to a confirmation page that says thanks for subscription, then next click on same link will get you the update profile page since you are already subscribed so there's nothing to confirm.

Hope it makes sense.
Can you just explain this? Is this per list or per account?
I have created a brand new list on an account with multiple lists. A subscriber might well already be on another list within that account, but not on the newly created list. Yet they get the Profile page instead of the Subscribe Confirm page upon verification.

I don't think that's a good solution and the user treatment should be unique to each LIST.
This is exactly how it works by default. If you get a different behavior, check your list settings.
I've looked in the known places and compared it with a group where it works correctly and couldn't see a setting that could cause this or that's different.
Sorry to bug you But this is still not working - I'm sure it's my own mistake somewhere, but all new subscriber get thrown to the "Profile" page after confirming from the double opt-in email. It's a real issue now. I cannot find the place where I went wrong.
I just tried this and I cannot replicate it, it just doesn't happen in a standard install.
Maybe open a support ticket and point us to the list that's causing this and also include customer access so we can take a look.