This answer covers key labels and descriptions with a heavier focus on popular music styles, as opposed to classical music:
I think the handiest way to lump all of that information together is usually "starts in X" rather than "is in X", when pieces of music have multiple keys throughout. Obviously if there is one main key (jazz standards which modulate in certain sections but return back, for example "Beyond the Sea", would fit this bill), then "this piece is in X" makes sense. But this question is asking specifically about songs and music where that is not the case.
If there are multiple keys and no clear primary key with which to label the piece, it is probably best practice to use a more descriptive statement. Here are some situational examples of what I mean by that, since every song will be different:
The Beatles' song "Let It Be" is in the key of C major. (This is not up for debate.)
"Enter Sandman" by Metallica is in the key of E minor. (The parts in F# minor lead strongly back down to E minor.)
Beyonce's "Love On Top" is in C major. (Many modulations up a half-step at the end, but still primarily C major - the ending is clearly a repeated variation of the main part of the song, so C has a convincing claim to being the key of the song.)
"Layla" by Eric Clapton is in D minor, but drops everything down a half-step to C# minor in the verses. (It makes sense to say that D minor is the primary key, since it is only the verses that temporarily shift.)
Frank Sinatra's recording of "Fly Me to the Moon" can be viewed in C major or in its relative key A minor. (There is a continuous blur between the two relative keys.)
"God Only Knows" by the Beach Boys has E major competing with A major to make a weak and fluid tonal center. (Though E is likely the most viable candidate, the song weakens the key by constantly flirting with other keys such as A major and avoiding strong resolutions to the E chords.)
"Bohemian Rhapsody" by Queen begins by tonicizing the key of Bb major and modulates to various other key centers. (It is not feasible to pick a single key to label the entire song with, but since it starts in one clear key, this can be used to communicate the necessary information.)
"Giant Steps" by John Coltrane cycles between the keys of B major, Eb major, and G major. (It is useless to single out any one of the three key centers.)
Schoenberg Piano Concerto Op. 42 is atonal. (Defies any tonal logic, and no key could make a claim to represent this composition.)
The concept of a key should not be applied to "4'33"" by John Cage. (Assigning a key signature to silence is pointless.)
Some might disagree with some of the descriptions I gave, but this is fairly subjective, and my point is that each case will be different and require different methods to communicate the necessary key-related information. There are many different ways that multiple keys could be present in a piece of music, and they must be described situationally as a result. It is not always going to be possible to state unequivocally that "X key is the key of this song".
Generally, the most important purpose for giving one single key for a song is to get every musician on the same page as to what to play, and in cases with multiple keys present, as long as one key is given for one agreed-upon moment in the music, every other part of the music can be deduced with some transposition. The trouble with that often lies in two places: either it is not obvious what the key at a particular moment is, or there is no obvious moment to select as representing the key of the entire piece, or potentially both. Thus, occasionally music may require a more detailed description of its varying tonalities in order to communicate unambiguously what to play.