> I always thought this was simply shorthand for something like: "I own shares #1034852, #1092647, and #2986246".

Nope, (private) shares are almost always fungible and not numbered/serialized in any way.

In fact most shares are just a name followed by a number of shares in an excel spreadsheet or Google sheet at best, if not simply a declaration in founding documents.

> People sometimes own 0.25 shares of a company, but I thought that this was a convenience invented by stock brokers like Fidelity and Robinhood.

Again, no. Fractional shares are very normal. Any time shares are created (via dilution, additional investment, etc.) some existing grants and holdings are likely to become fractional. Usually fractional shares are calculated to the third or fourth decimal place.

Many shareholders in the forementioned spreadsheet will have their share amount tied to a % calculation which will inevitably resolve to a fractional share amount.

Say a share is worth $500, are you okay with $500 rounding errors? I wouldn't be. With 4 decimals the maximum rounding error becomes an immaterial 5 cents.