It does NOT need to be Apple Products. I have paired Sony noise canceling headphones to my Apple Watch. And I have paired 2 very old (like 15 years old) Bluetooth headphones to my Apple Watch.
Unless there is something really funky about the Bluetooth headphones, you should be able to pair them with your Apple Watch.
There is ABSOLUTELY NO reason you have to buy Apple Bluetooth headphones to pair with your Apple Watch!
As to why they do not play music via the Apple Watch speakers, it is my feeling that driving the speaker would result in drawing too much power and drain the very small Apple Watch battery too quickly. The Low Energy Bluetooth is much more battery friendly, and you get to use the batteries in the Bluetooth headphones to vibrate air.