No: "like cotton" is neither a preposition phrase nor a modifier.
“Like” is here an adjective (meaning “resemble”), thus “like cotton” is an adjective phrase.
The “like” phrase is predicative complement of “feel”, a verb of sensory perception.
Note that the preposition “like” occurs as head of an adjunct, as in “Like his father, Ed became a teacher”.
If "like" were an adjective, as opposed to a prepositional phrase, in sentences like the one in question, it's strange that it can't be intensified like other adjectives:
My cat's fur feels very soft.
My cat's fur feels so soft.
*[strike]
My cat's fur feels very like cotton.[/strike]
*[strike]
My cat's fur feels so like cotton.[/strike]
On the other hand, "like cotton" can, like many prepositional phrases, be modified (or specified) by "just":
*[strike]
My cat's fur feels just soft.[/strike]
My cat's fur feels just like cotton.
Thus, I can't see the wisdom in dogmatically asserting, without any argument, that "like cotton" is an adjective phrase rather than a prepositional phrase.
The same applies to “His house is like a barn”, where “like a barn” is again an adjective phrase functioning as predicative complement of “be”.
Considering that prepositional phrases appear as complements to "be" (e.g., "His house is by the lake"), the ability for "like cotton" to function thus is not a good reason for thinking it is an adjective phrase rather than a prepositional phrase.
“Like” is here an adjective (meaning “resemble”)
"Resemble" is an interesting choice of synonym for an adjective, insofar as "resemble" is a verb, not an adjective. If "like" were an adjective in "My cat's fur is like cotton," the closest synonym would be "similar to": "My cat's fur feels similar to cotton."
However, that is not what the sentence "My cat's fur feels like cotton" means. The sentence means "My cat's fur feels similarly to how cotton feels."