Legal Question in Real Estate Law in California

Basically, no. A newborn infant which hasn't even been given a name can aquire ownership of land by inheritance, for example. You should be able to complete a purchase and record a deed without showing anyone any identification.

Your problems in having no identification would arise if you wanted to get title insurance, to borrow, or to sell land. You probably couldn't get a deed notarized, for example, which is a prerequisite to recording one (when you are the seller).

Note that if you have fingers and thumbs, their prints are a form of identification, useful for some purposes but certainly not all.

And just as a further caution, it is a felony to record a false deed, which could include a deed which substitutes a fake name to hide the identity of the true owner, for an improper purpose.