By the dictionary definition of "fake," CNN is correct to say that "the ad is false." With few exceptions (Stephen Glass, Jayson Blair), mainstream news outlets do not fabricate stories. They might display shades of bias or publish errors requiring corrections, but that does not make them fake.
Trump, however, has tried (largely successfully) to turn "fake news" into a catchall slur for reporting he doesn't like.
Observe: In an email about the ad, his campaign said it is "calling out the mainstream media for peddling fake news and not reporting on the fact that President Trump is making America great again."
By Trump's standard, news is fake if it does not promote the subjective view that he is "making America great again."
Trump has aimed the "fake news" pejorative at CNN more often than at any other outlet. Meanwhile, CNN President Jeff Zucker has warned that "the perception of Donald Trump in capitals around the world is shaped, in many ways, by CNN. Continuing to have an adversarial relationship with that network is a mistake."
It is easy to understand why CNN would ban Trump's insulting ad from its airwaves. How is a bit more complex. Here is the relevant Federal Communications Commission rule:
No cable television system is required to permit the use of its facilities by any legally qualified candidate for public office, but if any system shall permit any such candidate to use its facilities, it shall afford equal opportunities to all other candidates for that office to use such facilities. Such system shall have no power of censorship over the material broadcast by any such candidate.
The rules governing candidates and outside groups are different. A network is free to accept or reject super PAC ads, as it sees fit, but has "no power of censorship" over candidates.
The only way to get out of airing an ad like Trump's is to refuse to air any candidate's ad. Remember: We're talking about candidates for president in 2020. Trump doesn't have opponents yet - not any serious ones, anyway, and certainly none buying ad time.
If, for example, Martin O'Malley were already in the race and if CNN had been airing his commercials, the network would have had no choice but to show Trump's, too. For now, however, CNN can turn down Trump because it has not said yes to any other 2020 presidential candidate.
If Trump takes his "fake news" message into his re-election bid, he can probably compel CNN and other networks to air it - unless they are willing to give up ad revenue from all candidates.