Translation:The doctor drinks beer exclusively on Friday.
Yes it's even more common, the meaning will be similar but אך ורק has more emphasis
I think that's wrong because of the "but" -- that would be something more like הרופא שותה בירה אבל רק ביום שישי. Your version is a statement (the doctor drinks beer) followed by a qualifier (but only under the condition that it's Friday), whereas the given sentence is just a statement of the doctor's beer-drinking rules. They have approximately the same meaning, but it's a different sentence.
Is there a cultural or religious reason behind this? Or it's just a random sentence?
It's probably both. There is a religious reason, but it's probably just random
How is "on Fridays" accepted for ביום שישי .
Shouldn't "on Fridays" be only בימי שישי ?