Print shop saying color is wrong when testing in Photoshop

So y'all are presuming there is never a case where Photoshop should be used to inspect a PDF, but I think there actually might be, and without clarification from the OP we don't really know what's going on.

Consider that the PDF being opened may not be the final output from InDesign, but rather a PDF created in Photoshop that has included spot channels and perhaps the printer is verifying that the image was produced correctly. I can see where such an image, with differnt channels, might still be used for correct output if the spots were remapped using Ink Manager.

Look for a different print. PDFs shall not be opened in Photoshop for printing. Other printers will be happy to supply these security settings or even better.

and if apples were apples (say the file was rendered in Photoshop as Lab) the Picker should "guess" the same, but apples aren't apples anymore.

To illustrate Brad’s post, if your printer Imports your PDF as Lab Color, the PS Color Picker will correctly identify Pantone Spots from the same Book. Here my InDesign to PDF colors are from the Pantone + Solid Coated library and Imported into Photoshop as Lab Color:

The PS Color Picker set to the same PANTONE + Solid Coated Book correctly identifies the Pantone numbers

To expand on a few others' questons:

1) Is the intent for the Pantone to be used as a spot colour?

2) Does the spot colour appear properly as separate ink in Acrobat > Print Production > Output Preview

3) If so, what does the Output show for the areas with supposed Blending? You would have received a warning that retaining spot colours in that scenario is problematic.

4) If your intent is NOT to print spot colours, and everything is converting eventually to CMYK for print, why would it be even necessary to colour pick a Pantone??

Yes, your printer is making no sense. Why are they even doing this?

As for the reason they are getting a different Pantone, the colour picker "guesses" at a Pantone colour (from the selected library) based on the closest match in Lab values it sees with the eyedropper. Now, yes, the Pantone you picked has a specified Lab value in its swatch, and if apples were apples (say the file was rendered in Photoshop as Lab) the Picker should "guess" the same, but apples aren't apples anymore. Your Pantone has been rendered into either RGB (whether it be sRGB (most likely) or AdobeRGB), or worse CMYK. Hence the NEW measured values of Lab you would get in the Picker are completely different.

Is the print output of the Pantone color going to be Spot or Process?

The Pantone Solid Ink Libraries are defined as Spot colors. If you open a PDF with spot colors in Photoshop, the spot color would get converted to the chosen process color mode. If they choose RGB or CMYK Color in the Import PDF dialog, the color’s conversion values and appearance would depend on Photoshop’s Color Settings, which could be anything. To get the color to import with no appearance change they would have to set the Import PDF Mode to Lab Color.

The printer should be using AcrobatPro to determine what the actual color space and profile of the colors are.

Is the Pantone in ID an actual spot color for a separate printing plate? 

What settings did you use to make the PDF?

If the Pantone color swatch is applied to an object, but then blended with another, the result will be an approximation; not the honest Pantone color.

If the other person eyedroppers the color in PS (why?) they will see an approximation, not the real Pantone.

This PDF that is opened in Photoshop, was it created in Photoshop?

What sort of "color blending?"

they are opening the PDF in photoshop

Really? Find another printer!

and then color picking the pantone, and it's showing up as a different pantone for them, than what is in the file in indesign.

I am not sure to understand.

So, you used Pantone colors in your InDesign file, export it as PDF and now the colors are wrong? 

Can you share the PDF? And post a screenshot of the InDesign swatch panel?