Edit: Solved. I had to specifically choose headset from Windows' prompt when plugging in. Thresher isn't Synapse compatible neither so registering it is pointless.
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In 2019, I bought a Razer THRESHER Tournament Edition headset on Amazon. Brand new. For some reasons it worked well on my old computer without extra measures.
This morning, I came to plug the headset into my new computer I received last week and it wasn't being recognized. I don't have any "RAZER headset driver" among my devices in Device Manager neither (should there be?). As such, I don't have any audio.
I launched Synapse and realized the headset weren't among my registered devices. I guess it didn't cross my mind I had to register a headset back then for it to work properly?... I assumed I could just plug'n'play.
So I went to register it but despite writing the proper serial ID and product number (RZ04-02350100-R3U1), the website refuses to register it.
Tried to contact support, wrote my serial number on their website and I was told by their system:
Your product is no longer covered by Razer's Limited Warranty.
Product: Razer Thresher Tournament Edition
Your product is out of limited warranty and ineligible for free support.
So... What now?