How do I view color libraries within the color swatch panel like I can in Photoshop?

If you wish to view the same Pantone color library in Illustrator that you have shown in your Photoshop screen shot, you could open the Swatches panel in Illustrator.
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• In the bottom left corner of the Swatches panel is a panel button labelled "Swatch Libraries".
• When you click that button a flyout menu appears.
• If you choose "Color Books", then you can choose the same library (such as Pantone+ Coated in your screenshot) from the submenu that appears.
• Once that library opens, I recommend clicking the panel options flyout menu button (in the upper right corner of the library) and changing to one of the list views so that more information is displayed for each color.
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The colors are displayed as swatch libraries and appear in a different fashion than you see them in Photoshop but you should be able to make the transition.
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Hope that helps. :-)

It is different in Illustrator.

When you have selected an object with a color you can find the nearest Pantone color with Edit > Edit Colors > Recolor Artwork

Choose a 1 color job and the Pantone Library.

To find the closest spot color (of any system) get a printed swatches book and compare.

There's no way that color management and automatic conversion will be able to do that for you.

But because you insist:

Use Edit > Edit colors > Recolor artwork to convert any selected vector artwork into a chosen Pantone color space. Make sure color management is set up accordingly.

Or set the color guide panel to a Pantone library and then select a color in the colors or swatches panel.