This is not strange at all. But depends on the transmitter, transmit power level, receiver, and the how close the two antenna's are situated.
Electronic circuits are linear only over a certain finite range. Expose them to too high a power level, and they respond in their non-linear range. A non-linear circuit response (saturation, clipping, etc.) can convert a clean narrowband signal in a "dirty" wide-band spectrum. Most cheap SDR receivers have very little, if any, front-end filtering, so even signals far from the tuned band on display, if strong enough, can easily cause overloading into the non-linear range, and splatter all over a spectrum display.
If the transmitter and receiver (and their antenna systems) are far enough apart, path loss can attenuate a strong RF signal down into the receiver's linear range. But too close, and this won't happen.
There are receivers (SDR and otherwise) that are designed to work even in the presence of strong nearby signals (on different frequencies) by including more (front-end, IF "roofing", etc.) filtering, but these receivers are usually magnitudes more expensive than an RTL-SDR.
In addition, many cheap handheld transceivers barely (if at all) meet various regulatory requirements for spurious emissions. So your handheld could be generating an easily visible amount of out-of-band or wide-band RF noise, even to a non-overloaded spectrum analyzer.
So some names might be "receiver front-end overload", "insufficient blocking dynamic range", "transmitter spurious emissions", and others.