As I've said, I'm pro immigration, but this argument is a total strawman. It's true that we're a mongrel nation, having been repeatedly re-invaded over a period of a thousand or so years - but go and ask the Britons or the Saxons what they thought of their invaders. If the point of the analogy is that this ethnic amalgamation has been occuring throughout our country's history so it occuring today isn't a problem, it's one that would lead many to land on the UKIP side (ie "The Saxon's all but destroyed the Britons - why would I be happy that this is happening now?")
The point was the majority of what we perceive to be 'British culture' is actually stuff we took or borrowed from other nations, or stuff that other nations brought to us when their inhabitants moved here. The Royal Family are seen as British as they come, and they're from a line of Germans, with the Queens children themselves being half-Greek. Led Zeppelin, the Stones, Black Sabbath and the Beatles are probably our biggest musical legacy, and they all got started off aping blues numbers from African American musicians. Queen are celebrated as one of the most successful British bands ever, but Freddie Mercury himself was of Indian descent and born in Zanzibar. Tolkien is one of our biggest figures in literary fiction, and he was born and grew up in South Africa. Fish and chips originated in with Jewish immigrants from Portugal and Spain, and tea is something we picked up from China and India.
This idea that Britain has some form of cultural identity that stands alone and separate from the rest of the world, something we need to protect from the dirty foreigners, is wrong. The entire history of our country is pretty much defined by how we integrated elements of other cultures, and made them 'British'. There is no part of Britain's identity that stands apart from the rest of the world, because we have always been influenced by the rest of the world in one form or another. When Napoleon was taking power in Europe, we engaged in a decades long war to get rid of him. When the Nazis were steamrolling countries left and right, we dug our heels in and held the buggers off until America came to help. The Middle East is how it is now partly because of how we divided up countries in the early 20th Century. India is now one of the major economic powers in the world, because we finally stopped fucking around and gave them the independence they wanted.
Our entire country is defined by how we have acted with other countries, and what we have approbated from them. Sometimes we have done that well, sometimes we have done that very, very badly (China would certainly be in a very different place now if we had never kicked off the opium Wars with them), but good or ill, we have defined ourself by what we give and take from other countries. There is no part of British identity that stands like an ivory tower, separate from the rest of the world. There is no singular part of British identity that has to be defended and kept sacrosanct from foreigners. We have defined ourselves by those foreigners time and time again, and it is specifically that multi-cultural, melting pot approach that is the one thing that really counts as the British identity.