A guy writes an email to me saying: Don't be shy and talk to me.

A guy writes an email to me saying: Don’t be shy, talk to me.

I am interesting in knowing how would you understand the action “writing” here.
Does this sentence mean that the action “writing” of the guy is a habitual one?

No, I would assume the writer is using the present narrative form to bring his story to life - originally an authorial technique employed by authors and journalists who wanted to bring as much ‘life’/ ‘feeling of something happening now’ to the story for the reader as possible, it is a technique which is becoming more and more prevalent in everyday use.