Always check unusual displays with U8g2lib first. Oliver supports many different chips and many different displays.
I can't find the SSD1315 datasheet. Solomon are not very good with replying to requests.
Perhaps a reader has a link.
I suspect that the User registers will be identical to SSD1306
But some System registers may need different values. (these are set once in oled.begin() and never changed)
Without the genuine SSD1315 datasheet it is only guesswork.
David.
Edit. Found the datasheet here