I looked into this issue and got confirmation that Spotify doesn't support old versions of Firefox. Sorry about that!
In order to use the web player in Firefox ESR release, you will have to use an user agent spoofer. You can use Mozilla's own internal method of user agent spoofing, or you can use an addon which will remember on what site you want to use a different user agent, such as this one, or when you don't want to tinker with Firefox configuration (which can look daunting).
Have a nice day!
SebastyRock Star 16
Help others find this answer and click "Accept as Solution".
If you appreciate my answer, maybe give me a Like.
Generally speaking, yes. However, it should be noted that some web applications including popular SaaS like G Suite and Office 365 only support the latest Firefox release. If your organization is using one of those applications, please consider deploying the latest release instead of ESR to meet the requirements.
It's quite possible the web player is one of such websites that support a later version of Firefox, for technical reasons brought out by the dev response I linked here earlier. :)
Can you try again now? Firefox ESR 60.4.0 should be able to use the Web Player as long as it has the required module for listening to encrypted content. If this module isn't present only the first ten seconds of every track will be playable and updating to the lastest Firefox version should fix it.