A play, also called a drama or a comedy, is defined by American Heritage Dictionary as "A literary work written for performance on the stage; a drama." The literary characteristics of a play are that there is no narrator, except in special instances like Antigone, Our Town and The Glass Menagerie. The characters engage in dialogue. It is the dialogue itself that explains events, develops the characters and the plot, and moves the plot from conflict to resolution. By this definition and by these characteristics, Marlowe's Doctor Faustus is indeed a play.
Let's look at some of the particulars. Doctor Faustus is in fact one of the special instances in which a play does have a narrator of sorts. Marlowe included a Greek Chorus as is present in Antigone and other Greek plays. The Chorus introduces and concludes the play; gives critical background information; provides a concluding statement; adds explanation and information as needed:
CHORUS. Learned Faustus,
To know the secrets of astronomy112
Graven in the book of Jove's high firmament,
Did mount himself to scale Olympus' top,
Being seated in a chariot burning bright,
Drawn by the strength of yoky dragons' necks.
The characters engage in direct dialogue with each other, and it is their dialogue and their soliloquies, like Faustus's opening soliloquy, that reveal character traits, character conflicts, plot conflicts, background, complications, rising action, falling action, and the resolution--with the Chorus adding comments and information as needed. Here is an example of how dialogue between Faustus and Mephistophilis reveals plot development and conflict in Doctor Faustus:
FAUSTUS. But, leaving off this, let me have a wife,95
The fairest maid in Germany;
For I am wanton and lascivious,
And cannot live without a wife.
MEPHIST. How! a wife!
I prithee, Faustus, talk not of a wife.
FAUSTUS. Nay, sweet Mephistophilis, fetch me one, for I will have
MEPHIST. Well, thou wilt have one? Sit there till I come: I'll
fetch thee a wife in the devil's name.