I work as a precinct election official in Ohio.
- The stub is there for in-person voting, and ONLY to be removed by the voter just prior to feeding the ballot into the optical scanner (where the ballot is tabulated and stored in a internal memory drive).
- The stub is collected by a precinct official and put into a envelope ("Stub A Envelope"). That envelope is sealed at the end of the day during the closing procedures, and packed for storage at the county elections site.
- If there is an issue with ballots at that precinct and an audit is needed, election officials can match the voted paper ballots (also stored, separately) and the stubs in a separate envelope, to audit that the paper ballot was official. A ballot without the matching Stub would be cause to be checked if ballot was fraudulent (for instance, check ballot number against the voter registration, check for duplicate or out-of-sequence numbers.
Ballots submitted other than in-person on election day have the stub remaining. This includes absentee (either mail-in or in-person early voting), provisional ballots, curbside voting. Before those ballots are scanned for counting, they procedures for each of those types of ballots would also include removing that stub and storing separately from the ballot for possible auditing.
As for your concern about matching the ballot number to your name, that is the case for ALL ways for submitted ballots, even in-person. During in-person voting, the ballot number is recorded when the ballot is given to the voter during the voter check-in process (when the voter id is checked, and voter signs the poll book). This ballot number recording has existed as long as I have been working at precinct elections, during the current optical scan ballots, the previous, computer touch-screen, punch cards, and lever actuator poll booths. It has always been recorded. Auditing may involve contacting you to verify the submitted ballot.