I see some people coming to airplanes with lot of plastic bags, often containing something bought from airport shops. Don't these count towards carry-on baggage allowance? Could I bring more carry-on stuff from home by using plastic bags?
It depends on the airline.
Jet Blue makes no mention of airport shop items in their carry-on policy, but the United Airlines policy states that "a limited amount of duty-free merchandise or food purchased in the airport" doesn't count against the carry-on limit and the Delta policy doesn't count "food or drink purchased after clearing the security checkpoint."
Carry-on items from home would count against the carry-on limit in these situations.
I am not aware of a specific rule to normal plastic bags but almost anything you carry onboard is considered luggage and rules apply to them, it is just that agents do not take a one normal sized plastic bag (eg. shopping plastic bags) seriously and overlook it except in cases where the bag is really huge. I always see people carry plastic bags along with their trolleys/hand luggage during boardings with no complaints from cabin crew, and as long as they made it to the airplane then surely the ground agents also had no problem with that.
Regarding plastic bags from duty free, they usually are bought after security checks, even if not they are totally ignored, they can not make people buy stuff then stop them from taking it onboard!. Plus, there are limitations on how much you can buy and/or how much you can spend in duty free shops, either from the destination country or origin country, hence you don't see people with oversized/many plastic bags from the duty free.