The warning is not fatal and you don't need to worry about it. Happens to everyone, all the time.
Your color shift happens because the X-1a preset converts your CMYK mismatches to SWOP preserving the color numbers rather than preserving the color appearance.
Really does not make any sense does it?
Right it doesn't—you'll get it with all of the default presets. If you make a change, something like toggle Pages and Spreads, it will go away.
Also when a image which does not have an embedded profile is present in the PDF its color in the PDF does not come close to matching PhotoShop
What's the color mode of the image without the embedded profile? An RGB or Lab image would get converted to CMYK (X-1a does not allow RGB) and colors could shift if they are not in the document's CMYK gamut.
If you would use RGB images instead of CMYK images you would not have a problem with wrong CMYK output profile or any CMYK conversion. Preserve numbers will change the color appearance and not the ink numbers, this setting helps to a workflow with RGB images placed.
No, it doesn't quite work that way. RGB images or vector artwork are converted to CMYK for absolutely all PDF/X-1a output, converting from the RGB artwork's RGB color space (either by virtue of the ICC color space of placed content or the document's default RGB color space) to the CMYK color space designated by the output intent profile. The preserve numbers option is totally irrelevant for any conversions from either RGB or LAB to the output CMYK color space. There's nothing to be preserved in a thee colorant to four colorant conversion. The preserve number option only deals with CMYK to C'M'Y'K' conversions in which case if preserve numbers is specified, there isn't really a conversion for any CMYK color, but rather a retagging to the specified CMYK color space.
And yes, that warning message is generally misleading if not absolutely wrong and if your color management is properly setup, can be safely ignored.
Dov, I know. But what I meant is exactly that what you wrote and that is why I recommended, that they should use RGB images because these are converted in X-1a to the selected output profile.
Assuming the description of the color settings is accurate, placing RGB wouldn't solve the OP's complaint of changed color in the PDF. If the Color Settings are sync'd to US SWOP and the InDesign document has the same profile assigned, an untagged CMYK image wouldn't change in appearance because in Photoshop it's being managed by the US SWOP working space, and in InDesign it's manged by the same document US SWOP assignment. The only explanation for a PDF/X-1a color change would be placing RGB, Lab, or grayscale with Overprint turned off and then exporting, which forces the color into the SWOP CMYK gamut.
If the problem image is RGB, the solution would be to turn on Proof Colors in Photoshop and set Proof Setup to US SWOP, or convert the image to US SWOP.