Who was buried under a false name and why?

Anne Catherick was buried under a false name, so Lady Glyde could be placed in an assylum without anyone's knowledge.

There is the fact of the funeral at Limmeridge, and there is the assertion of the inscription on the tomb. That is the case you want to overthrow. What evidence have you to support the declaration on your side that the person who died and was buried was not Lady Glyde? Let us run through the main points of your statement and see what they are worth. Miss Halcombe goes to a certain private Asylum, and there sees a certain female patient. It is known that a woman named Anne Catherick, and bearing an extraordinary personal resemblance to Lady Glyde, escaped from the Asylum; it is known that the person received there last July was received as Anne Catherick brought back; it is known that the gentleman who brought her back warned Mr. Fairlie that it was part of her insanity to be bent on personating his dead niece; and it is known that she did repeatedly declare herself in the Asylum (where no one believed her) to be Lady Glyde. These are all facts.