The PlayStation Gold headset is probably the only wireless headset which is directly compatible with PS3, PS4, PS5, Android, PC, Switch (Personally tested) Get hold of them before they are super rare. :
How to Connect Bluetooth Headphones to a PS4
Genuine Sony Wireless Stereo Headset for PS3 & PS4
Amazon.com: PDP LVL50 Wireless Power Stereo Gaming Headset with Noise Cancelling Microphone: Compatible with PS5/PS4/PS3 Console/PC Windows 10, Comfortable Lightweight Headphones, (Black Camo/Camouflage) : Patio, Lawn & Garden