Well, eticket numbers are 13 or 14 digits long -- a 3-digit airline code, a 4-digit form number, a 6-digit serial number, and optionally 1 checksum digit.

All United tickets are going to carry the "016" airline code.

That leaves 10 digits of unique namespace for United-issued tickets. If they're all useable, that's one hundred million eticket numbers that United can ever have outstanding.
(Edit: Uhhh, it was kind of a long night with the kids. Ten billion, sorry.)

Tickets are valid for one year from date of issue.

United carried 158 million "passengers measured by each flight segment flown" in 2018 (most recent annual Form 10-K, filed Feb 2019).

Even if only 80% of passengers are flying on 016 stock and if 100% of passengers are on a 4-segment journey covered by 1 ticket, seems like they must be pretty close to issuing 100 million eticket numbers per year? Reuse seems inevitable within a century.

I'm kind of surprised that you managed to look up an eticket without a last name -- how did you manage that?