One is incorrect because the relative pronoun who refers to the singular antecedent 'animal'.
We sometimes imagine cats to be a sort of independent (animal who lead) mysterious lives of their own.
-We sometimes imagine cats to be INDEPENDENT ANIMALS who lead mysterious lives of their own.
Two is correct.
-We sometimes imagine cats to be a sort of independent animal (who) leads a mysterious life of its own.
A few other points to consider.
a sort of = something in the nature of, approximating the qualities of but not having all the qualities of
-He is a sort of Napoleon.
sort of (Colloquial) = adverb. somewhat, to a moderate degree
-He is sort of crazy. We imagine cats to be sort of an independent animal. (sort of independent animals)
* Teacher (Brad) is correct. The relative pronoun should be 'that' or 'which' when referring to animals. Pets with names or animals that are characters in stories are often exceptions to this rule.
Ex. My dog Bim, who(m) everyone knows as Beemer, is the world's best dog.
I think sentence number two is basically okay.
The problem arises when you use the linking verb "to be" and switch from plural to singular. CATS to be A SORT of
This switch is allowed in the English language but to avoid confusion you can use the singular form on both sides of the linking verb 'to be".
We sometimes imagine the cat to be a sort of independent animal that leads a mysterious life of its own.