I don't know for sure how those horrible-seeming devices work--but even if the dog's side of the system is meant to be only a receiver, then if that receiver circuit works anything like a regular AM or FM radio receiver, it almost certainly radiates some amount of RF energy as the received signal is converted to an intermediate frequency (traditionally just above 10 MHz). And since the designs of nearly all consumer electronic devices in the past 20 years or so have been penny-pinched to infinity and beyond--and since the modern FCC believes in trusting the free market to enforce standards--you can pretty well bet that the manufacturer doesn't give a rat's patootie about what problems their equipment may be causing your equipment (or anyone else's but their own).
So yes, it's plausible that the collar could interfere with many types of audio equipment, not just microphones. But microphone signals are at lower voltage levels than many other audio signals, so they're inherently more vulnerable. And the capsule end of a microphone amplifier operates at very, very high impedance, so stray coupling can be fiendishly hard to prevent.
The RF spectrum has gotten much, much busier in the past 20 years. Most audio equipment designed before that time, or newer equipment that imitates older designs, simply can't cope. Back in the day the concern was mainly about cable shielding. which definitely helps to keep AM broadcasts out. But with the increasing use of higher frequency carriers and the use of very aggressive digital modulation of those carriers, PLUS the increasing likelihood of very close proximity between an RF source and audio equipment is relatively new; in the past, transmitters (or RF leakage sources) didn't typically move around in the middle of public gatherings, walk across your cables, or stop and sniff at them.
Engineers of past generations mostly weren't trained in the area of "EMC" ("electromagnetic compatibility") at all; most audio design textbooks until recently didn't deal with the issue. And technical managers and marketing executives need to care about EMC enough to spend real money and time on it, but many do not--in part because they're generally older and less exposed to present realities, and in part because the realities they care about the most are on the corporate balance sheet, and they really want to save that last handful of cents of profit per unit sold.
All of which adds up to, I don't know, but certainly maybe.
--best regards and I'll stop now.