Yes ExxonMobil does make the Toyota 0W20. Of course it contains a different additive pack from their own Mobil 1 AFE 0W20. There is some threads over on bobistheoilguy.com comparing the two.I've read many places that Mobil makes Toyota's oil. They have had a relationship since the 1950s.
That is the engine lubrication plate from a 1958 Toyopet/Toyota Crown.