An MP3 isn't a single FT or DCT of the entire waveform either. The format breaks the stream up into frames, which are then processed.
So when engineers say "it's a Fourier Transform" they mean:
"It's a Fourier Transform the same way that we use it in practice which is to first break up the audio in the time domain and then transform each chunk into the frequency domain using a Fourier Transform or Discrete Cosine Transform for analysis of that chunk of audio."
But it's quicker to just say "It's a Fourier Transform".