Many Taiwanese are proud of who their are today. As many Taiwanese words are not Southern Min any more but borrowed Japanese words. Taiwanese love Sushi, Udon, Soba, Monga, Anime, Karaoke, Japanese industry and Japanese cultural influence. The reality is that because Japan, in the post war years, has been the most infuencial asian culture and business that bring Asian pride to Taiwanese. A role no doubt will be taken over by China soon.
"Taiwanese" is in the category of South Min Dialect as Beijing dialect, Sichuan dialect, and Northeastern dialect are in the category of Mandarin. In Taiwan, han chinese which accounts for 98% of local populaton speaks two dialects apart from official standard Mandarin, SouthMin and Hakka. South Min has different names in different chinese-speaking regions. In taiwan, it is called "taiwanese", in Singapore and Malaysia, it is called "hokkien" or "fokkien", which means "福建, in china, south min also many varients. the most influential one is Amoi (Xiamen dialect), taiwanese dialect is almost identical to Amoi.
BTW,Sushi,Anime, Karaoke are nothing less popular in china