I found it's spoken quite commonly but if I'm a speaker, I'd rather try to say something like "can you make a copy of this for me", that is using "for me". The original sentence looks like it's having "me" as a direct object. How is this correct and what grammar is applied to it?
Can you make me a copy of this?
Can you make a copy of this for me?
Both forms are fine and are similar.
These two sentences have the respective constructions:
a. modal verb + S + V + NP Dative + NP
b. modal verb + S + V + NP + PP Dative
The noun phrases are the direct objects of their respective verbs. The indirect objects are either in the NP form, like NP Dative, or embedded in prepositional phrases, like PP Dative.
wikipedia explains dative shift.
[D]ative shift refers to a pattern in which the subcategorization of a verb can take on two alternating forms, the oblique dative form or the double object construction form. In the oblique dative (OD) form, the verb takes a noun phrase (NP) and a dative prepositional phrase (PP)....
(1) John gave [NP a book ] [PP.DATIVE to Mary ].
In the double object construction (DOC) form the verb takes two noun phrases...with the dative argument preceding the other argument.
(2) John gave [NP.DATIVE Mary ] [NP a book ].
In the title, you have asked for the meaning of make. make means produce as in 'produce a copy of this'.
Not all transitive verbs accept indirect objects.
Wikipedia explains monotransitive and ditransitive verbs.
Transitive verbs can be classified by the number of objects they require. Verbs that accept only two arguments, a subject and a single direct object, are monotransitive. Verbs that accept two objects, a direct object and an indirect object, are ditransitive.
More explanation on transitivity and examples on use of monotransitive verbs, among others, are found here.
The cat bit me!
He broke the toothpick.