Ah, thank you for the clarification, that is a nuance I completely missed.
A follow-up question since you know more about this than I do: if some number of people each buy a fraction of a share, who gets to vote that share, nobody? Does it become effectively a non-voting share?
A fractional share is a new financial instrument operated by the seller of the fractional share.
If you take the example of Robinhood, here's the relevant section[1].
> "I understand that fractional shares within My Account (i) are unrecognized, unmarketable, and illiquid outside the Robinhood platform".
Robinhood remain the owner of the share and retain the voting rights. Robinhood's terms describe how they distribute dividends and voting "rights", but this is a commercial agreement between Robinhood and the fractional share buyers, and doesn't invoke the legislation surrounding share ownership.
When it comes to fractional shares as a consumer-facing product, I imagine it would be effectively a non-voting share, yes.
However the vote could be exercised by whatever entity actually owns the whole share. That would be entirely a decision for the people who chose to divide up the resource. Presumably BH will only accept a single binary vote per share, so it would be up to the owners of the fractional share to agree (or not).
They also provide indivisable benefits: the right to vote, and the right to inspect company documents.
The solution to owning "a thousandth of a share" is a stock-split, where the smallest fraction is still one share.