I would say that all is is functioning as a determiner here, home as an adverb.

English Teacher
In the following sentence, which parts of speech are the words all and home: The dog and cat ran all the way home.
I would say that all is is functioning as a determiner here, home as an adverb.
That's a good question!
I agree about "home" You could take "all the way" out and it would be clear that "home" answers "where".
Regarding "all the way"???? Think of a couple thinking about going "all the way". Could it be replaced by "completely" or "entirely". Then you might call it a "phrasal adverb". But, really, at the moment, I can't say.
If "the way" is what they ran, as in "I walked the way to Santiago de Compostela, " "way" could be considered a noun acting as the direct object and could be modified by the (in this case) adjective answering the question "how much of".
An interesting question.
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