Hi spaceman,
Given our history & ties with Britain through the years the whole question of who was what, & when is quite complicated, & is further complicated by the various immigration acts & laws that have been introduced over the years. Changes in Dominions & the Commonwealth makes things a bit trickier, as does the perception of nationality & citizenship, along with how different categories of people are described & how their even their rights & related laws have also changed. Ireland falls into a special category in UK citizenship/immigration laws.
So, the easiest way to ‘categorise’ someone in terms of what they were at any given time is to see what category they fall into now.
When your father was born he was a British subject, though he would now be categorised as an Irish citizen, given the fact that he did not make a claim to remain a British subject. If he had done this & remained living in ROI, he would have the right to a British passport, but he would not be described as a British citizen on it, so would not be able to use it, for instance on the US visa waiver programme. If he had become a British subject & subsequently lived in the UK for five years he could then have become a naturalised British citizen without taking an oath of allegiance to the queen. But, if he were alive now he could still apply to become a British subject. There are other rules & exceptions giving people special rights, such as if he had served in the British Army etc. The rules about British subject status changed in 1949 and again in 1983.
The same applies to you, seeing as you were born before 1949 – you could apply to be a British subject, but, again that would not make you a British citizen. With EU laws you could become a naturalised British citizen anyway after living there for five years. The category of British subject will become extinct over time after all existing subjects or those who have the right to be, are dead.