Several universities founded in Mainland during the period of Nationalist China, or even earlier, had been reintroduced in Taiwan since mid 50s.  Those schools continue to operate in either side of the strait now under same old name in both Chinese and English, eg. Tsinghua, Sun Yat-sen University; or under different English name, eg. Chiao Tung vs. Jiao Tong;  or vice versa, eg.  Soochow (東吳 vs. 蘇州). 

The day before yesterday, with several former classmates, I went to National Chi Nan University(暨南大學) located at Puli, Nantuo, a very large and beautiful university that is rare in Taiwan.  The school was founded somewhere at Jiangsu Porvince in late Qing Dynasty, originally for overseas Chinese and foreign students.  Now the university in Mainland has been restored in Guangdong under the same Chinese name, but difference English name from that of Taiwan: Jinan University.  

The names of those universities are not that important.  What really matters is their academic reputation and achievements.  All of those schools in China Mainland are prestigious ones: Tsinghua, Jiao Tong, and Sun Yat-sen are all the members of 985 Engineering(top 38 universities in China), while Soochow and Jinan are among 211 Project(100 key universities in 21st century, including those of 985 Engineering).  They all have acquired much higher standing in the world ranking systems than their namesake counterparts in Taiwan.  So Taiwan should have done a whole lot better than ever; otherwise, the difference will be surely wider and wider.   

PS  The other day I read a piece of news on LINE in which the principal of Chengchi University advised his students of not seeking further studying in Peking University and Tsinghua University because they are "low class" schools.  What the hell, are you nuts, Mr Principal?  According to many ranking systems, both Tsinghua and Peking are around top 20 in the world.  How about Chengchi?  Far from reaching top 500!  For crying out loud, man!