Do they just show an Illustrator icon, or do they really open in Illustrator? If the open they are definitely NOT ID files.
It sounds like you may be mistaking PDF's for InDesign files.
They show the illustrator icon and then open in illustrator, but the original file was done and saved in InDesign
Definitely sounds like you're opening PDFs in Illustrator.
InDesign files do not open in Illustrator.
What operating system are you on?
Show the file extension - it should be .indd not .pdf.