So, I’ve found a solution at least for my setup and a probable bug. I’ve got HAOS in a KVM VM and Frigate elsewhere in a docker container. I had the Frigate addon pointed to Frigate over port 8971 (TLS was disabled in my config per the Frigate manual to allow HA to talk to Frigate without SSL issues).
Under these circumstances, I got the same error as you, and the docker logs for Frigate were indicating HTTP status 401 (Unauthorized) when trying to play the clips and recordings.
Figuring this might be the source of the problem (and a bug), I enabled port 5000 in my docker-compose.yml and pointed the Frigate add-on to it.
This worked for me and I can now play the clips/recordings from HA. I also added an IPTables rule to the host holding the Frigate Docker:
iptables -I FORWARD -p tcp ! -s MY_HOMEASSISTANT_IP --dport 5000 -j DROP
… to deny everything else from being able to contact the unauthenticated 5000 port other than my HAOS install (the MY_HOMEASSISTANT_IP is a placeholder for your HAOS’s IP).
Hope this helps someone else out!