Or does this antenna have a frequency tolerance?
This is closest, but the antenna, itself, does not actually have a “frequency tolerance” (more usually called a bandwidth). Rather, the antenna's properties, including impedance/SWR, radiation pattern, and losses, vary with frequency, and the range of frequencies over which you can transmit are mainly determined by your transmitter's ability to cope with varying impedance of the antenna — i.e. how high a SWR it can transmit into.
If you read the documentation for an antenna, it may specify a frequency range, but that will always mean something like “the SWR will be no higher than [x] between frequency [a] and frequency [b]”. If you choose a higher acceptable SWR, the range becomes wider.
However, this is more relevant for tuning range than for the ability to transmit FM — typical antennas will have usable SWR over a much wider frequency range than any individual signal's bandwidth or FM deviation. Consider this: if they didn't, then you couldn't change your transmit carrier frequency without modifying the antenna, since a meaningful change of transmit frequency is one which is wider than an individual signal's bandwidth!
(Some types of antennas amateurs use, such as the “magnetic loop” antenna, have much narrower bandwidths to the point of requiring retuning — changing the antenna impedance to be favorable at the desired frequency — for frequency changes within a band. These antennas have their own tuning mechanisms designed for frequent use, unlike dipole or vertical antennas where the tuning mechanism is usually “cut to the correct length”.)