Many green grocery shops were just that. But, as is pointed out, some did fish and poultry. I think the fish and poultry side of a business was not as regular as greengrocery. There was always a supply of greengrocery and fruit, which in days past, did vary according to season. However poultry seems to have been more festive in nature - nowhere as plentiful all year around as now - likewise fish could be scarce during winters especially when there were storms at sea and expensive. Deep sea trawlers fared better but the day trawler, mainly coastal, was more affected and those buyers who could charge more for their fish usually got first choice. Rabbits, of course, which were generally cheap, were plentiful in earlier times.
People did tend to grow more of their own vegetables and keep fowl in the past which most certainly helped the family budget.