Apart from a number of people named Ray who perform as Santa Claus, the word would normally be spelled "rey" in Spanish, meaning "king"; the plural is "reyes" as in Point Reyes.
Searching the Oakland Tribune archives, the first references for "Santa Ray" are indeed for a racehorse, beginning in 1905. The horse was owned by J. F. Clifford. 1 References to the street name don't appear until a few years later in 1908. 3
In 1910, which sections of the street were called Santa Ray was adjusted a bit. 2
Other streets in that same area, which was originally called East Piedmont Heights and Piedmont Heights could also be named after racehorses.
Calmar Avenue and Rosal Avenue
Lerida Avenue (now Balfour)