How many Queens regnant of the UK bear the name Elizabeth?
The answer is of course 1, even if Elizabeth is termed the Second.
The fact of the matter is that Elizabeth I was Queen of England. In 1707, England and Scotland combined into the United Kingdom of Great Britain, after a hundred years of on-off sharing a monarch or commonwealth government. Ireland then joined in 1801, combining the treble crowns of James I (or VI of Scotland). This kingdom only had one Queen Elizabeth.
This numbering wasn't an issue initially as all the monarchs had names that had not been in use by the earlier kingdoms, so bear the correct number (George I, II, III, IV), up until William who would be III for Scotland but IV for England. This was the height of the Scottish being seen as 'Northern British', so they just ignored it and called him William IV. The issue actually then came up after Victoria repopularised Scottishness with her son Edward. In the end, based on the example of William IV, it was decided to number the monarch according to the highest number of any of the constituent countries. Thus Elizabeth is the II, although Scotland never had such a queen before. Interestingly, this means that if a royal decides to rule as James for instance, he would be James VIII - which would look strange in lists of English kings, where they would jump from II to VIII.
As an aside: Edward has its own problems with numbering. There are three English King Edwards that antedate King Edward I - Edward the Elder, Edward the Martyr and Edward the Confessor.
Initially kings weren't numbered. The English only started to do so when King Edward I was suceeded by an Edward, who in turn was also followed by one. Being the height of the middle ages, they ignored the pre-conquest kings as they were only thinking of their own time. Thereafter Edward has always been numbered wrong, although some historians add 'since the Norman conquest' to correct it.