As history is seldom written to tell the absolute truth, (if such a creature exist at all), the history or historicity, (defining the latter as a "quality of being historically authentic"), of the languages spoken in the vast continent of what is now politically defined as "China" would include or rather needed to include elements from areas beyond the academic insularity of liquistics.
So to answer the question "Do many Taiwanese self identify as Mandarin speakers and if not, why?" would require a mini thesis of sorts simply because it involves a clash of the divergent politics of national identity between the Mainland and Taiwan, (perhaps even saying "...between the Mainland and Taiwan" is itself controversial)
Added into the witches brew is the "contribution" from Western scholars and diplomats who did just as much to shape what the Chinese themselves call what they speak.
For a start, the word "mandarin" to mean the "mandarin" as a language is itself a Western invention, and having gained so much scholastic mileage that it gained a life of its own in Western writings.
It is therefore not surprising that the Taiwanese, (i.e. those who were not exposed to any substantial amount of Western writings), were confused. After all why would the literarily meticulous Chinese call their language after a mere senior court official? The Westerners who had the initial contacts with the 17th / 18th Chinese did not know what to call the language spoken in the Chinese court, and so simply called it the language of the senior court official -- "mandarin" (Like the way we say, "just google it") Heck, even the word "China" is a Western invention.
And so why would the Taiwanese, past, present or future, "self identify as Mandarin speakers", even if the "Chinese" that they spoke is more or less the same as mandarin as we know it?
As for 普通话 "PuTongHua", there is nothing 普通, (ordinary / common), about it. It is based on what we would loosely refer to as "mandarin chinese", which just 50 years ago would be unintelligible to, my opinion, about 70% of the population who historically spoke their own dialects. It was a tool, (ala Shi Huang Ti), to engineer a politically united China, and the ordinary / common part is just so as not to step on any dialectical toes in the politically fragile environment of post-revolutionary China.