The possessive is acceptable in your examples, in which case they mean
His memory was like a baby's memory.
Ian's car was like his car.
However, English accepts the so-called double possessive in which this elision doesn't apply:
She is a friend of my sister's.
John is a fan of hers.
It's redundant since of, -'s, and possessive pronouns all signify the genitive, but language isn't necessarily logical. The form has its use in distinguishing
A picture of my family
which is a portrait of your relatives and
a picture of my family's
which is a picture your family owns.
We generally reserve the doubled possessive for human beings, like babies and sisters, but not otherwise. No one says
the founder of the country's.