Is this really a democracy
Yes. It's really a democracy because it has universal adult suffrage and governments are regularly voted in and voted out in reasonably fair elections.
Let's put it this way, if the line dividing real democracies from not-real democracies is drawn such as not to allow the UK to qualify, then there are very few real democracies in the world. Certainly not the USA with its electoral college!
One can certainly defend a prescriptivist view of what constitutes proper democracy. The trouble is, everyone's prescription is different. Here is an incomplete list of types of countries claimed to fall short of being truly democratic on various grounds:
countries with a ceremonial monarchy, or with an electoral college, or without proportional representation, or with the wrong sort of proportional representation, or without proper constitutional safeguards for individual rights, or with constitutional safeguards for individual rights (after all, don't they temper the pure majoritarian glory of Athenian democracy?), or without a vigorous opposition press, or with a slanderous and irresponsible opposition press, or which have signed up to supranational bodies outside democratic control, or which have signed up to international treaties ditto, or which make use of protected seats for women or ethnic minorities, or where the if the prime minister is one ethnic group the president is conventionally always another, or where unlimited campaign donations are allowed, or disallowed, or where significant numbers question the legitimacy of the polity holding the vote, or where voting is compulsory, or where non-citizens cannot vote, or where they can, or where there is (or is not) a second chamber or supreme court that can block popular legislation…
Many items on that list have a far bigger practical effect than the ceremonial pretence that the person with a very uncomfortable-looking metal hat actually gives permission for anything to happen. Yet if we disallowed all those countries from being democracies we would have to think up some another word to mean "those countries in which, however imperfectly, the people are ultimately in charge."