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The simple answer is no. Especially not between Sennheiser and Shure. My advice would always be to stick with the manufacturers recommendations, especially if this wireless mic is key to your show. But as with everything in life there are some exceptions. Wisycom make transmitters and receivers that can be mixed with other brands products. This is especially useful for bands that do world tours, because the Wisycom IEM transmitters have a much wider frequency band then Shure or Sennheiser transmitters, but still support the artists favourite flavour of belt pack. Shure PSM1000 transmitters can be picked up by UR receivers if they are in the same band. I have used this many times doing car shows when you need a stable remote audio connection, but can not have any cables on the ground. The lower end Sennheiser stuff is somewhat interchangeable as well. An example being a 300 series handheld can be used with a 100 series receiver as long as they are in the same frequency band. More on reddit.com

Look into the Sennheiser wireless mics. G4 or AVX-ME2. Can get a system for that price second hand. More on reddit.com

Just to piggyback on u/MathmoKiwi excellent feedback - in your use case (congested RF, long distance, use of handheld/plug-on transmitters, etc.) also carefully consider your Antenna needs and receiver diversity. The challenge handheld mics present is how often the antenna changes orientation as the mic is moved around. For reporting use, this is why you often see a reporter-style mic used with a short xlr lead connected to a body pack transmitter/receiver that's antenna is always oriented up to match the orientation of the receiver's antennas. Reporter body packs will also often provide a talkback receiver so the station can speak to the reporter (tangent) While this is important for any transmitter, when using handheld wireless mics or plug-on transmitters the FM wave changes its orientation corresponding to the orientation of the mic. For this reason using a true diversity receiver (two receiver modules for each receiver channel) will yield better results as you have the option to orient your antennas to match the orientation of the FM wave. The receiver will then seamlessly switch between receiver modules depending on which module is achieving the best signal to noise. Many professional wireless systems have some type of diversity (e.g. Antenna Diversity) but not all are True Diversity. The marketing material will gloss this over so read the specs carefully. So right inline with this is the need/cost for proper antennas and block-specific filters. LPDA shark-fin style receiver antennas with block-specific filters will work well in your situation as long as your not pointing them at the source of strong RF interference. Think of these like you would a hyper-cardioid shotgun mic that is most sensitive where it's pointed and rejects from sides/back. RFVenue also makes an interesting antenna called the Diversity Fin that is optimized for mics that change orientation with an LPDA fin antenna combined with omni antennas (two connections) that is optimized for mics that change position. I can't say strongly enough how beneficial block-specific (band-pass) filters are. These filters go between your antennas and receiver ensuring your receiver is only hearing frequencies within the range of your transmitters. In congested RF environments these will lower the noise floor, increasing the signal-to-noise ratio which is what good RF is all about. Regardless of the choice between a new digital or analog system as the same laws of radio frequency physics apply to both. Digital is not a silver bullet for RF and in most cases makes RF best practice even more important. Again, try to get beyond the marketing hype and really dig into what a digital system has to push over that RF carrier to fully appreciate what's needed. Lastly, if you have any budget left over (LOL) look into receiver/transmitter packages that provide a wide tuning range in order to take advantage additional new groups/channels that are less congested than 470-608 NA bands. In Canada we have some almost unused bands in the 900 range available which are a dream to use. Lav mics? I'm a big fan of DPA and Cos-11s. More on reddit.com