Which part of speech is "as" in each example of mine?

[1] He is [the same as the dog (is)].

In [1] "as" is a preposition and the expression "as the dog (is)" is thus a preposition phrase functioning as complement of the adjective "same". The verb "is" can be optionally added. The whole expression "the same as the dog (is)" is an adjective phrase functioning as predicative complement of the verb "be". Note that "the" is a dependent of "same".

[2] He is just like the dog.

[3] He is similar to the dog.

In [2] "like" is an adjective" and "just like the dog" is thus an adjective phrase functioning as predicative complement of "be".

In [3] "to the dog" is a preposition phrase functioning as complement of the adjective "similar". "Similar to the dog" is thus an adjective phrase functioning as predicative complement of "be".

Note that all three examples are comparative constructions.