From what I understand after investigating this issue, this happens with multiple TV types from Sony.
I got a new TV, Sony BRAVIA, 49XG8096, 4K Ultra HD.
The TV is cool, no issues with it at all.
My plan was to connect it with a Bluetooth headset too, so I can watch a movie without wires, even if my wife is sleeping next to me.
I first bought a JBL Tune 500 headset, just because that was on stock, and could not lose time searching for anything else. The headset was working fine, connecting with anything, but I couldn't get it to connect to the TV at all.
I looked up this issue and saw that there is a list of headphones that are on an Official list from Sony. I can't say I was happy, but I went back to the store, and changed the headphones to a Sony this time, model WH-CH500B.
According to the list, it should have worked.
Well, it didn't.
I investigated some more, and I found out that this is might be a software issue, with a very easy fix.
I went to the TV's Play Store, searched for Bluetooth, and installed the first 3rd party app that I found. The app connected the Sony Headset to the Sony TV in less then 5 seconds.
After this, I can just go to the TV settings, and the WH-CH500B is listed as paired, I can easily turn it on or off.
I am thinking that the JBL might have been working too if I used an app like this.
I am not sure why Sony cannot do this themselves, but I don't care anymore.