After 2 hours of frustration with different controllers and even failing pairing with ps3controller driver, I decided to try a USB BT adapter (disabled the internal adapter) and only that lead me to the root cause.
I had forgotten a similar USB BT adapter attached to my HTPC nearby and it still had PS3 driver running.
I noticed this when trying to pair sixaxis driver with this same kind adapter in PI. First I was happy that the pairing found the device and seemingly successfully paired with it but controls wasn't working after unplugging the USB. Still the DS3 light showed that it had connected "with something". Only at this point I recalled that I might have another dongle still attached in one of the several PCs running nearby.
It was weird that it did not pair with this HTPC dongle when I was trying to pair it with PI internal BT radio, but only when using the similar dongle. Also, the internal radio did not ever found the controller but neither did the controller connect to that other USB dongle.
Silly mistake but further more I didn't think I'd have another dongle around as these controllers have been used for years with a Nvidia Shield and the original PS3 controller without a problem and with that forgotten dongle attached to HTPC all the time.
I verified that the sixaxis driver now works both with Pi3 internal radio and with the external dongle.
Thanks and sorry for being dumb.
Edit: Just to add, I successfully paired DS3 controller and a non vibrating original sixaxis controller. Thought to mention as while searching for help I found some discussion about wondering if the older controller type would pair at all.