Quince is assigning the roles because he is the one who wrote the play, calling it "The Most Lamentable Comedy and Most Cruel Death of Pyramus and Thisbe." Because he is the playwright, he is allowed to assign the various parts. This suggests that Shakespeare had the same privilege.
While the subject of the play seems in questionable taste to celebrate a marriage, ending as it does with the tragic suicide of two young lovers, the other mechanicals, such as Bottom, respect Quince's abilities. Quince does appear to have thought carefully about who should be cast in what role and to have fashioned the play with certain people in mind.
The play is certainly based on Ovid's story of Pyramus and Thisbe in the Metamorphoses, showing that Quince is literate and reasonably well read in the classics. It would also be common to adapt a play in this way from another source.