It's meaningless to specify bandwidth alone for a high impedance passive voltage oscilloscope probe. In fact, they are already specified as system bandwidth.
A specific oscilloscope probe will only fit certain oscilloscope. Passive probe relay heavily on compensation to achieve target bandwidth, which is related to a certain front end circuit.
Here is how LeCroy said on their website:
Each passive probe is recommended for a certain oscilloscope, using the right passive probe with the right oscilloscope means that the probe will be properly compensated across the entire bandwidth. Using probes with a different oscilloscope will only let you compensate for low frequencies.
For most brand, you can find a probe compatibility list on their website or probe datasheet.
To my own experience, a keysight compatible 500 MHz probe will have strange frequency response above 50 MHz on a LeCroy, Tek, or Siglent.