TL; DR type of answer:
In short, MP3 is a lossy format that does distort the audio waveform in which the C64 data is stored, but just like you can still listen to morse code or music just fine on a noisy or distorted radio channel, the digital data signal on tape may be transferred and stored with audio equipmentand it can survive a lossy audio compression format, as long as the audio signal is not too distorted in the whole process for the C64 tape drive to pick up and restore the digital data signal within some margin of error.
A more in-depth answer:
MP3 is a lossy format, and while it does do damage to the signal waveform by removing frequency content to encode it smaller, the distorted signal waveform can still be good enough to be able to convey data. It depends on the MP3 encoder settings, obviously, but there will be other difficulties in the system of transferring tape data between C64 systems as MP3 audio files.
So first of all, when MP3 encoder has thrown away data, it cannot be filled in or recovered any more.
But this thrown away data is not C64 data bits, this is audio signal energy information that is thrown away.
Second, the audio energy that was not present to begin with in the original audio signal will not be encoded. For example, silence needs much less bits to encode than speech, music, or white noise.
Third, the big point which is also important, that the encoder will also keep information about energy or throw it away, by using a psychoacoustic model to determine what in the audio us humans can or cannot hear, and allocate more bits which need to be presented with higher fidelity than the parts that are not so important. So encoder does not care about what part of the audio waveform is important for transferring C64 data, the encoder only cares that us humans hear the C64 data similar enough before and after encoding.
Knowing that the C64 data on audio tape is basically represented as square wave signals that have high or low tone frequency (or rather, time between edges of the square waves) to represent bits of ones and zeroes, we can analyse how well it might pass MP3 encoding or get distorted. Simple test of generating 1 kHz square wave in an audio editor, exporting it as MP3 file at pretty low quality of 80-120 kbps, and importing it back reveals some distortion, but it genuinely looks good enough a square wave to work. There is slight peaking at the edges and the edges have some ringing, which is not surprising at all due to loss of high frequencies in the MP3 encoding.
So, transmitting C64 data over MP3 files is definitely doable.
The problems lie elsewhere than in the MP3 encoding.
The C64 data stream is a digital square wave signal, not an audio signal, even if contemporary home computers did use analogue audio signals to standard cassette recorders. C64 data is transmitted as digital square waves from C64 to tape drive (datassette), and the write head stores the edges of the fast digital square waveform to cassette tape, with fast magnetic transitions. So again, not audio. Playing the cassette back in an audio tape player that outputs analog audio will have bandwidth limited to audio frequencies, so the signal edges are slower. A real C64 datassette will directly convert the sharp magnetic transitions to digital square wave for sending it back to C64.
If the data tape is played in an audio tape player and fed into computer sound card for recording, it will be an audio waveform which is bandwidth limited square wave with some limited slew rate in the otherwise sharp transitions. But at least it can be recorded, stored as MP3 and distributed.
Playing the MP3 back to tape is also problematic. Since the computer plays already bandwidth limited square waves to tape recorder, the tape recorder will store the transitions of the square wave less sharply. Recording audio is more complex than just writing analog waveform directly as magnetic signal to tape. While many cheap devices did that, higher end recorders biased, or modulated, the audio with 100kHz sine wave, to allow for a better recording of the analog waveform. The recorders anyway limited the bandwidth, and the square wave transitions stored on tape are not very sharp, and the sharper they are, the better the tape works in real C64 datassette. As long as the square wave edges are reasonably sharp with high enough amplitude, it will be detected as a transition by the C64 datassette circuitry and will be output as fast clean square transition on the digital signal to the C64. But with too slow transitions of too low amplitude, the transition timing may get distorted or there may not be a transition detected at all.
And finally, the reason why lossy MP3 will not distort the waveform beyond recognition, is the fact that the signal is essentially just two different frequency square waves. And square waves just have energy at the base frequency and odd overtones. The more overtones that can be stored, the better looking the square wave is, but it is still limited by tape bandwidth. As tapes don't generally go above 15 kHz, and MP3 format starts also attenuating high frequencies after around 16 kHz, it means that for example 1 kHz square wave needs only 8 frequency peaks in an MP3 file. So the recording of C64 tape data has only energy at few sharp frequency peaks, and the rest of the frequency band is unused, so the MP3 encoder can use all the bits to encode the scarce energy content quite faithfully.