Does it sound weird if it was "The Person of the Year" with article "the"?
Not necessarily! The original name of the award is "Person of the Year," without "the." The speaker would have said "the Person of the Year" if it was introducing the name of the award, without the possessive "Time Magazine's" before it. Normally, the possessive form will not be followed with an article unless it's already in the name of the proper noun. Ex: The "Person of the Year" award goes to Greta Thunberg! Ex: The winner of the "Person of the Year" award is Greta Thunberg. Ex: The winner is Alejandro G. Iñárritu's "The Revenant!" (Possessive 's + the article is allowed here because the article is in the proper noun's name)