Another post on the AI forum:
https://community.adobe.com/t5/illustrator-discussions/importing-a-color-group-from-libraries/m-p/13749144

If you right click a color group (containing global colors) and select “Add Theme to Swatches”, it creates an empty folder in the Swatches panel. If you create and select an object (like a rectangle) and click on a color in the global color group, it does not fill the object, but adds a magenta Spot color to the Swatches panel.

A recent Illustrator forum topic:
https://community.adobe.com/t5/illustrator-discussions/cc-library-colour-swatches-breaking/m-p/13408316#M347042

Still not working. They don't get added twice, but that's it. You would expect a seamles, color managed exchange of colors (and other swatches) between CC apps...
This is the description of what is shown in the movie.
In Illustrator.
Create a new CMYK file.
Create a spot color and add it to a CC library. Close the document

Create a new CMYK file.
Just Double click a Spot Swatch in the Library, click OK.
Color gets added to the swatches as Spot (although the Color Mode has become CMYK).

Create a new CMYK file.
Draw a rectangle.
Click the Spot color in the CC library.
No color added to Swatches, color fill becomes Process.
After that the color in the Library has permanently changed.

Spots added from Illustrator show CMYK values in CC Library.
Spots added from InDesign show Lab values in CC Library (and become added to swatches as 100% magenta process color when double clicked in Illustrator).

Double click a Global CMYK Swatch in the Library.
Color gets added to the swatches as a 100% Magenta Spot color.

The bug was added to the backlog in 2019.

@Ashutosh, could we please get an update as to wether steps are beign taken to correct this?
From a print oriented profile, I feel this is a major brake to getting users on board with libraries.

This one: Spots become Process Magenta is still not solved:
https://illustrator.uservoice.com/forums/601447-illustrator-bugs/suggestions/34291627-pantone-solid-spot-colors-added-from-indesign-to-a?tracking_code=8ff2c6d23ad86db2b07d1c16c7e36e2d

Create a Pantone Spot color swatch in InDesign (I used 321 C for sentimental reasons) and add it to a CC Library.
Switch to Illustrator, double click the swatch in the CC Library to add it to the Swatches panel.
It becomes a Global Process Color CMYK Swatch as 100% Magenta instead of a green Spot color.
https://shared-assets.adobe.com/link/2dbbba8a-e688-486f-41ee-cba1d77530df

Almost solved, still not there yet. When you apply a global color from the CC library the color gets added to your illustrator document, but the name has changed to some hexadecimal value. Which, if you look at it, is also in the name when it gets added to the library. So why is the name of the color changing? And if it was just a processcolor (not global), the color isn't added to my next Illustrator-document if I just click on it when an object is selected (I have to right click in te library to add it to my document) . O yeah, by the way: the spot color from Illustrator has a CMYK value/alternative instead of Lab. With LAB you would have a better representation of the color, but illustrator doesn't store them as book colors. So still not quite there. When I add a Pantone color from InDesign the color has an RGBvalue as book color in Illustrator, not Lab anymore. Why do I feel that I do the research for Adobe?

>>> Is no one testing this before they give us the update? You drive me crazy (to say it politely).

Thanks Molly,
After reading Peter Maas comments, I went back and the color that behaved well first, lost its name and became a normal process color. It seems that it works the first couple of times after the global color is added but then changes to a non global not custom named color.

Hello everyone, thank you for your patience as we worked with the Illustrator team to fix this issue. With the latest 3.18 CC Libraries release, you can now add a global process color from the swatches panel to your libraries. Try it out today and feel free to provide additional feedback!

- Molly D (Creative Cloud Libraries team)

Still not fixed? Are we giving up? What's the use of a library when you can't use it. :(

I've spent nights over this problem just to find it's not me, it'sAdobe.

We're up to Illustrator v25.4 but you still can't add Global Process colours to your library.... This is a ridiculous issue, please fix it Adobe!

Still not fixed?

In Illustrator, it seems the workaround solution is to preselect the item (text, shape, etc.) to be assigned the colour from your Library, and THEN right-click the library spot colour and select "Add to colour swatches".
Pantone spot maintains it's spot assignment, and imports to swatches with saved name, and only one copy.
If you have nothing selected, and try to add a colour from your Library to swatches, it makes 2 copies of the same colour in CMYK values.
Hope that helps!

In Illustrator, it seems the workaround solution is to preselect the item (text, shape, etc.) to be assigned the colour from your Library, and THEN right-click the library spot colour and select "Add to colour swatches".
Pantone spot maintains it's spot assignment, and imports to swatches with saved name, and only one copy.
If you have nothing selected, and try to add a colour from your Library to swatches, it makes 2 copies of the same colour in CMYK values.
Hope that helps!

We need a fix! This bug becomes more confusing with Illustrator on the iPad - because when you take a library theme color and save it in you document swatches on the iPad it AUTOMATICALLY becomes global process. hmm

In fact any swatch you mix and save becomes global process on iPad Illustrator.
When you open your .aic from the iPad in Illustrator on the desktop, swatches stay global, process, yay yay!

But if you save any of those colors in the library, they become spot (sigh).

bump this...still a bug in March 2021!

+1
Really hope Adobe fixes what is seemingly a fundamental flaw.

+1. Still an issue three years later.

Would really love a solution to add Global Colors to my Library without them changing to a spot color.

Adobe, where is your response? Essential to fix this; design software for designers, this affects the design process.