TOPIC:

I recently started playing CB again and I see a lot of people transferring mail and passengers to their station in cities.

I can't really tell if this is allowed. Could someone clarify this rule for me?

Rule #4: No self-competition on primary industries is permitted. Also cheating to advance yourself ahead of the competition unfairly will not be tolerated.
Is this considered a primary industry?

Rule #7: Do not station spread with airports, docks, train or road vehicle stations to extend coverage area to additional resources including towns and cities unfairly.
You're not extending the train station here since you are using those road vehicle stations

Is this considered a primary industry?

Transferring like this is allowed. It's how you extend coverage over more of the town without station spreading.

You're not extending the train station here since you are using those road vehicle stations

Generally correct, station tiles you're using aren't usually considered station spreading. In your screenshot though, it would be. We generally allow road stations to extend max 2 tiles away from the train station, and the one in your screenshot seems to go 3 tiles out.