Green is the color of the Democratic Progressive Party (currently in power, and the historical anti-KMT (read: anti-authoritarian) party)[0]. Green has thus become the color of the Taiwanese independence movement (i.e. not "Republic of China", but "Republic of Taiwan")
Blue is the color of the KMT (former dictator party)[1], whose logo is still the top-left quadrant of the flag of Taiwan[2]. Historically, this party has claimed to be the politically legitimate government of all of China (as the "Republic of China" rather than the "People's Republic of China"), but lately is effectively an arm of the Communist Party of China.
It's a long and fascinating story that involves China firing missiles to affect the outcome of Taiwan's first democratic election, as well as the largest American military action in Asia since the Vietnam War.
In summary, the KMT upholds a "One-China Policy", which makes them a natural ally of the CCP against any Taiwan independence movements.
Blue is the color of the KMT (former dictator party)[1], whose logo is still the top-left quadrant of the flag of Taiwan[2]. Historically, this party has claimed to be the politically legitimate government of all of China (as the "Republic of China" rather than the "People's Republic of China"), but lately is effectively an arm of the Communist Party of China.
[0] https://en.wikipedia.org/wiki/Pan-Green_Coalition#History
[1] https://en.wikipedia.org/wiki/Pan-Blue_Coalition
[2] https://en.wikipedia.org/wiki/Flag_of_the_Republic_of_China