I have lived in Taiwan quite a while now and come to admire some brands/companies in Taiwan and associate some as being more 'Taiwanese' than others. For food I have to say Imei always has quality products, it runs it's owns stores in little towns all over the island and has been around since the 1930s. Ten Ren Cha's stores and drink outlets are top quality, drink and food wise...if they changed their marketing style for the West they could escape out of Chinatown
In electronics HTC has impressed me these last few years, it makes me a little bit proud that a Taiwanese heaquartered company can make world beating phones. They show the potential here of companies that are willing to incorporate foreign offices and design to become multinational. Acer and Asus are kind of there but not quite, Acer took the route of low margin quick revenue and Asus is for nerds and gamers and also took a 'cost-down' approach in the main. Giant makes cracking bikes as do many other bike companies and parts suppliers.
85C does what it does very well...good coffee and okay cake for a few bucks. They have a huge future in China.
Brands and companies that don't impress me here would be President/7-11/Family Mart which sells a lot of sweet crap in it's shops and too much plastic packaging, just bad for health AND the environment. Carrefour (although not Taiwanese) and RT Mart are very boring and sells a lot of processed stuff to me but Matsushita has a good mix (that being Japanese originally of course). I've tried La New shoes here, they weren't bad but a bit overpriced for what you get and the design was quite plain.