To elaborate on what others have posted, you're hearing a voting comparator shutting off a voting receiver after a transmission.
The voting receivers are placed over a specified coverage area to increase talk-in range of a system. When the receiver is sitting idle and not receiving a transmission, it sends a continuous 2175hz tone down the leased line, microwave or RF link.
That 2175hz tone tells the comparator to "ignore me." When a voting receiver does receive a transmission, the tone disappears and passes the received audio. When the transmission ends, it again sends the 2175hz tone.
The "plink" you're hearing is the period of time after a transmission has ended, but before the comparator has muted the site from voting.
Correct me if I'm wrong, Eric, but you can head multiple clicks depending on how many layers of comparators you're going through.
I miss that sound after leaving Seattle. They're still running an analog SZ system. However, several of the conventional systems here in LA use AstroTAC/DigiTACs. As an added bonus, they're all T-Band, so it's nice wideband audio.
LAFD has a conventional voted 800mhz analog system with voting receivers. They've turned off reverse burst for some reason, so you hear a squelch crash, then the "plink" from the voting comparator shutting off.
https://www.youtube.com/watch?v=YKdb7owvA7A
Bonus round:
JPL voters have their own characteristic "growl" at the end of transmissions:
https://www.youtube.com/watch?v=Unzf_zkw5do
And a BNSF rail dispatcher clicks after dekey:
https://www.youtube.com/watch?v=6B0ne24hlmo