Okay, thanks to process explorer, I figured it out.
When you first register, this is how it will look from now on. This is through the in-game client.
However, even if registration is not done through the browser, it still leaves chrome cookies on your linked osu account.
This is how it looks initially.
But once you finish the registration process, and make an account on the client, the cookies added are what identifies you and warns the client about the existence of another account in the machine. A cookie wipe will always be needed from now on, for multi accounters at least.
EDIT: Next release will be tomorrow. Make sure to UPDATE OSU!, or else it will not be compatible.
Changelog: uniqueID changer (for MBR partions).
Y'all can forget about the cookieless browser because registration is dealt with through the client in the newest update.
And yes, Cherry-... Client modification is necessary to avoid the behaviour described above. I'm looking into it.