I had a look at your original Illustrator file.
Pantone spot swatches, for many years now, are defined as LAB values.
In the original Illustrator file you sent (in your PDF), even though it is defined as a spot colour "Pantone warm Gray 1 C", someone has changed it from its LAB values to CMYK (0C 2M 3Y 6K). Whether this came from an old legacy file or was converted with inappropriate Color settings, I don't know. In any case, it's not a match for the current Pantone LAB swatch at all.

This creates a problem when you import it into another program that does use the current Pantone LAB swatches. So, when you bring the file into InDesign, it will map the Pantone to the same-named one in its own Library, which is LAB-based... with the values of approx. L84 a1 b4.
So, depending on your Color Settings, when you change these to CMYK at output, they will convert according to what ICC profiles you have in place. e.g. if you're using an old CMYK Profile like US Sheetfed Coated, you will probably get something along the line of 15C 13M 15Y 0K.
If you really want to match the SPECIFIC CMYK values of your original grey, I suggest you give it a new spot name in the Illustrator file (say, e.g. "Special Gray"), so when you bring it into InDesign, it will add that same colour into its Swatch Library.