Why do we use the objective case pronoun "me" in "He can watch me clean the car"?

He can watch me clean the car.

In this sentence, the pronoun me is used as the object of the verb watch. But isn't me also acting as the subject of the verb 'clean' and therefore should be I instead? Obviously 'he can watch me clean the car' sounds correct but why is it this way? Is there a rule that governs this?

Reopen note

This question has been linked to this question here:

• "Heard me [infinitive]" vs. "heard me [present participle]"

However, that question is clearly about whether to use an infinitive or a gerund-participle form after the verb hear. This question is as described above. It is about the function of the pronoun me in sentences like the one described.

"me clean the car" is an infinitive phrase as the object of the main verb. The subject of infinitives is always in the oblique case.

sbj: He
vrb: watched
        obj: me clean the car (infin)

Other examples of infinitive phrases as objects:

My mother made me take the trash out
She wanted us to come
We asked him to go home