Do many Taiwanese self identify as Mandarin speakers and if not, why?
Mandarin originally refers to the official language of Qing

By the end of the last dynasty in 19th century, the government KMT released some standard of 国语(national language), which is based on the 官话(mandarin) of the Qing dynasty. 
When KMT went to Taiwan, they took the standard there. On the other side, PRC made the standard 普通话(common language), which is based on the KMT's 国语.

So when you talk about the mandarin of Taiwan, it is 国语, in mainland, it is 普通话, however, 国语 and 普通话 are almost same, they are both based on mandarin of Qing and the dialect around Beijing.

Before I visited the English sites, I don't know mandarin too, I remember when I got the explaining from dictionaries, I was like wtf, 清朝官话? So what we speak is called the official language of Qing, fk me.  I don't like the word, it always reminds me of the Manchu barbarians.

edit:

Some rumors say mandarin is from 满大人(man da ren), some other proofs show that it is used before the Qing dynasty.

This is written by Matteo Ricci in the Ming dynasty in Latin, which means high-ranking official used by Portuguese.