Traditional markets are a big part of Asian culture. In Japan, they're not as popular, though Ameyoko is still going strong, certainly catering to tourists but also selling fresh foods including fish. And of course, the Tsukiji Fish Market is a very active part of the restaurant business. Tokyo especially has seen the small shops move into purpose-built actual structures (with offices, more shops, or housing on top), and many of the same shops gathering in one neighborhood.
But in Taiwan and Korea, I saw plenty of markets selling, well, everything. One evening on my first trip to Korea, I found myself wandering around a market area that led to another market which had tons of food and kitchen products.
I wish I knew which market I went to, partly because I'd like to go back again sometime. But there are plenty of traditional markets in Seoul and around Korea. And I hear the spicy food is really spicy when you go local.