What do people mean by “personal identity”?

I have always been weirded out by the concept of personal identity. So called “identity philosophy.” I do not understand the importance of personal identity examples, the “ship of Theseus.” People say that if you replace one plank, does it cease to be the same ship? If you replace entire planks of your memory with false ones, do you cease to be the same person? But I don’t understand how the answer matters, where the distinction is supposed to lie. It seems that a hypothetical ship would be the same ship regardless of one or two planks - it is still a ship, and still a part of the world, and still having an effect, still floating, still in the story of the world. The material constitution of a ship would make it a ship. People do not have a material constitution. They are organisms. I don’t understand how identity is supposed to matter in this case. If a person has a memory altered and is unable to tell, is it still them? Why does it matter if the answer is yes or no? So what if it is or isn’t them? Could anyone draw a line on the number of planks that must be replaced before theseus’s ship is no longer his? Do we therefore conclude that it is a matter of degree and not a binary question? I just can’t see how the question of whether a person is the same can matter - they are still a person.