How does a plane's weather radar ignore returns from other weather radars?
&quot;Radar to radar interference&quot; / &quot;Mutual radar interference&quot; is an ongoing research topic, though much of the current research (based on my cursory Googling) seems focused on interference between radars on self-driving cars.  That leads me to believe a certain degree of radar-to-radar interference is currently just tolerated.
The link shared by randomhead gives us some insight into how maritime radars mitigate mutual interference, both directly and indirectly.  The same techniques should be applicable to aircraft weather radars, which tend to also be X-Band and also be pulsing radars.
Directly

The interference rejection capability of the radar essentially relies on timing differences between the radars.  A pulse radar sends out many pulses, separated by around 1 millisecond.  Microsecond-resolution timing between the transmission of the pulse and the receipt of its echo is used to determine range to a target.
Interfering radar pulses will arrive somewhat randomly on that microsecond timescale, but actual target echoes (e.g. from weather) will be correlated across multiple pulses.  The radar can be made to only acknowledge a return as real if enough echoes arrive.  Related topics here seem like &quot;m-out-of-n&quot; detection, &quot;binary integration&quot;, &quot;coincidence detection&quot;, &quot;dual threshold detection&quot;, and &quot;double threshold detection&quot;.
Activating the &quot;interference rejection&quot; mode on a radar (or having it always on, which I suspect is the case on many aviation weather radars) will cause a radar to reject certain weak real returns (more false negatives).  Luckily, clouds full of precipitation tend to return radar fairly well.


Indirectly

A different section of the same link gives instructions for tuning radar.  This implies that there is room in the band for more than one transmitter to be operating simultaneously, though it may not be organized formally into channels.  This is reinforced later in the description of the Search and Rescue Transponder (SART) scanning the radar band for pulses to reply to, which wouldn't be necessary if they all operated on the same frequency.
So, automatically or manually, weather radars should also be able to deconflict by frequency separation.  This will, of course, run into limitations as the number of transmitters grows, but at least it reduces the amount of work the Interference Rejection mode has to do.

If the radar can change properties of its pulse, such as the pulse length or frequency modulation, it can potentially use those properties to distinguish between its pulses and pulses from other transmitters.  The link doesn't really support that being used to reduce interference in maritime navigation radar, though, and I kind of doubt it's used for such in aviation weather radar either.  I think I've heard of schemes like that being used for more warfare-oriented radars but that's even further outside my own expertise.