Well kids can get rectal and vaginal strep just from more casual contact so it hardly seems a stretch to think that adults could as well, albeit the environment is a bit different.
Certainly Herpes Simplex Type 1 (HSV1), usually an oral organism, is found as a genital infection now, having travelled by that means - and HSV2 is in oral areas by the reverse.
Of course there is the old joke about two doctors talking about there interesting cases of the day:
Dr 1: I never saw it before! Would you believe it? Gonorrhea of the Big Toe!
Dr. 2: Meh. I just had a lady in with athlete’s vagina!
It’s certainly possible- cite(last paragraph on that page of a Google book).
And of course, there’s always Group B strep which is a deadly lil’ bacteria for newborns passing though the vaginal canal, and all sorts of other pathogens that can enter the body from the Vaginal route, but it’s not the most common causes certainly. But if you’re asking “is it possible” then the quick and dirty answer would be “yes, it’s possible.”
Actually no to the thrush - Candida (the yeast that is thrush) is is both areas all the time, infections depend upon something happening that allows the yeast to overgrow, such as antibiotic use, stress, other illness such as diabetes, etc.