The grammar here is not particularly straight forward. So, for anyone just starting Latin, "mihi" is not always "my" (which is more accurately the possessive adjective "meus, mea, meum")
"mihi" is the dative case of "ego" (I). So, while the given translation is perfectly good, if you are trying to understand it literally it is something like "The name to me is Marcus."
They're not interchangeable any more than "he" and "him" are interchangeable. "Marcus" is the nominative and "Marce" is the vocative.
Here is a plain-English overview of what the cases are and how they work:
Latin cases, in English
Adjectives must agree in gender, number, and case with the nouns they modify, but they have their own declensions. Sometimes you get lucky and the adjective just happens to follow the same declension as the noun, but that is not a guarantee.