This isn't true. Spartiate families owned land (a kleros) that was inherited, the kleroi weren't equal, and they became increasingly less equal over the history of Sparta. Many families lost their Spartiate status because they couldn't keep up with the wealth requirements of their class.

Spartan writings had a decline narrative, where they said all Spartiates were more equal in the past, but there isn't any evidence this was actually the case. Just more ur-fascists worshipping a mythologized past.

Here's a historian blogging about Spartan equality