I am also having the same issue. Tried in different networks (home and work), tried in incognito mode, tried to clean cookies and history. Nothing works. I really need Spotify to run the Web Player on Firefox.
Thanks for your suggestion. This didn't work for me unfortunately.
Just cleared history for all options (cookies, cache, active logins etc.) and for time range everything. Closed firefox completely and re-opened, still getting the same message 'This browser doesn't support Spotify Web Player. Switch browsers or download Spotify for your desktop. '
As mentioned before, changing the User Agent to a later firefox and spotify works again without any issues.
So simply swapping User agent from
Mozilla/5.0 (Windows NT 6.1; WOW64; rv:60.0) Gecko/20100101 Firefox/60.0 (won't load)
Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:64.0) Gecko/20100101 Firefox/64.0 (loads and works perfectly)
Just confirmed on the firefox site that 60.4.0esr is the latest release in the Extended Support Release family. It's current, maintained and supported and it's use is common place for user desktops in many corporations.
Like others here have already said I don't have unrestricted choice of browsers on my work machine so I hope spotify aren't blocking / restricting it and that this is just an oversight.
For anyone interested I used the 'Browser-Agent Switcher' addon available through firefox addons to swap between User Agent versions.
My home machine is running Firefox 64.0.2. When i visit open.spotify.com it works fine, when i set the User Agent to the details of the ESR release and then try to access open.spotify.com It returns 'This browser doesn't support Spotify Web Player.'