Q: So does this phone have oleophobic coating?
A: The screen protector which comes with the phone does, the glass screen itself doesn't. (Still needs to be confirmed)
Q: Can I use this phone without a screen protector?
A: Yes you can, but it's suggested that you use a screen protector (The phone comes with a screen protector already applied on the phone)
That is incorrect, that is the shatter-proof laminate. Xperia Z does bundle both front and rear screen protectors in the box, which the user meant to apply themselves.
Here is what Sony support said about the shatter-proof laminate sheet:
The scratch resistant screen is covered in a non-scratch resistant anti-shatter laminate sheet that will reduce the likelihood of the screen cracking or shattering. This is NOT the top layer intended as the laminate sheet is very easily scratched and attracts lots of dust and fingerprints. You are then supposed to put a screen protector over this laminate sheet which will do a much better job of resisting dirt, fingerprints and scratches than the laminate sheet. One is included in the Xperia packaging in unbranded models. The preinstalled laminate sheet is glued down. It can come off but it will leave a glue residue on the screen that is not easy to get off. The second layer screen protector included with unbranded models just bonds through natural static adhesion onto the laminate sheet rather than a glue.
The laminate sheet can not be replaced on its own after removal from the screen. The laminate sheet is bonded to the screen with heat as part of the production process in order to properly fuse to the glass and give it its anti-shatter properties. The laminate sheet is not sold on its own as a part and the only way to replace a damaged or removed laminate sheet is to replace the entire front screen glass panel assembly. As such Sony highly recommends you immediately install the screen protector included with direct from Sony unbranded Xperia models or purchase a commercially available screen protector for carrier branded units.
Due to this laminate, it is therefore highly unlikely that the glass surface has any oleophobic treatment.
Also, from user tests, it is highly imperative one apply screen protector on top of the laminate, as it is easily scratched: