There does not seem to be any way to change the order of the profile tabs on the user page. Maybe weights or similar on the profile types that would allow a drag & drop reordering? Or a simple field when editing the profile type that is normally blank.
Needs review
1.0
Code
Not all content is available!
It's likely this issue predates Contrib.social: some issue and comment data are missing.
Here's a patch to make this functionality available.
It adds weights to profile types and makes them draggable, similar to vocabularies and taxonomy terms in core.