With classic search engine indexing you can find and remove exact matches from the index, but with neural networks it's harder to make sure you removed every representation of a specific information from the parameters. For example you remove somehow the exact username-name from the model parameters ( that doesn't seems to hard at first) but then it may still return the information if somebody ask the model differently.

So if you try to remove the information from a neural network model then it can still have it in different forms you may not even think of, for example in language models the same thing described with different words.

And on the other hand removing one thing may affect the models performance on other unrelated things too.