You don't own the share you lend as in it's no longer your property, but you own a new contract claim against the brokerage house whom you lent to. As with any loan, this is newly created financial property. It acts and is priced like the original share, plus additional compensation for the risk (payment for share lending).

It's not the same as company issuing a new share since there is no voting right and there is no SIPC insurance. But in all other ways, financially, it's as if the brokerage created a new share. Real share owners can lend again, so one real share can leverage into any number of outstanding synthetic shares. The only thing holding leverage in line is the margin requirement, and how many people who lent real shares want their loans back.

Right so one share does not become two and "sythenic shares" is as much a thing as "synthetic money" that is money you loaned out.