It sounds like you're pulling the photos into a folder using Copy. This creates a duplicate, and yes, that duplicate can have a different name. But do you really want to create a duplicate? If the photos are already on the computer, you might want to Move it to your new chosen location, or Add it at the existing location, rather than creating a duplicate file (with or without a different filename).
That's not a problem for Lightroom, because they'd have different file paths. It's generally recommended that you use unique filenames for a multitude of reasons, not least because if the photos go "missing" (numerous causes, such as moving files outside of LR, drive letter changing, etc.), then it's far easier to reconnect them correctly if they have unique names.
Lightroom renames the real files, so you can never have a file that has a different name on disk than it has in Lightroom. What probably happened is that you copied the files to another location and renamed them in the process, but now you are looking again at the originals on disk. It doesn't matter that the originals aren't renamed, because Lightroom does not do anything with the originals. They are not in your catalog.
