Thomas Aquinas deals with this question in Third Part, Question 3, Article 4: "Can one Person assume without another?"
He argues that the assumption of a human nature implies two things in Article 2:
in the word assumption two things are signified--to wit, the principle and the term of the action. Now to be the principle of the assumption belongs to the Divine Nature in itself, because the assumption took place by Its power; but to be the term of the assumption does not belong to the Divine Nature in itself, but by reason of the Person in Whom It is considered to be.
This distinction between the "principle" (the Divine Nature) and the "term" (associated with the Second Person of the Trinity) then plays out in Article 4:
Now the act of assumption proceeds from the Divine power, which is common to the three Persons, but the term of the assumption is a Person, as stated above (Article 2). Hence what has to do with action in the assumption is common to the three Persons; but what pertains to the nature of term belongs to one Person in such a manner as not to belong to another; for the three Persons caused the human nature to be united to the one Person of the Son.
A Reformed theologian, Louis Berkhof, similarly writes:
It was not the triune God but the second person of the Trinity that assumed human nature. For that reason it is better to say that the Word became flesh than that God became man. At the same time we should remember that each of the divine persons was active in the incarnation. (ST, 3.2.1.B.1.a)