up to our necks vs up to his neck

up to our necks and up to his neck both are adjectives.

Word	Noun	Adjective	Verb	Adverb
up to our necks	No	Yes	No	No
up to his neck	No	Yes	No	No

As adjectives, up to our necks and up to his neck are synonyms defined as:

Other synonyms of up to our necks include neck-deep, up to her neck, up to my neck, up to their necks, up to your neck.

up to our necks (adjective)	up to his neck (adjective)
deeply involved	deeply involved