Store Data on Users Google Account

That is not possible, it's just not within the goals and purposes of an(y) authentication provider.

Have you thought of:

• How would Google keep "your" piece of data secure and separate from other sites that would try to do the same for that user?
• How would Google deal with the possibly Gigabytes of data that it would have to store (per user, for multiple sites combined) within days after this feature would be introduced?

The way this is usually done is by creating a database / datastore, that you then maintain and make backups of (and possibly pay for) in which you store data for each user, using some User ID as a "key".