Illustrator item in PANTONE P Process Black U shows as composite black in InDesign

As mentioned by others, there is no need to ever use Pantone Black in a document instead of using the Black swatch that already exists, and certainly not in a project that is intended to print in process.

Regardless, unless you are actually using the colours for a spot colour job, you should NOT used any of the libraries with U at the end. Why? The different between a C (Coated) colour and a U (uncoated) colour is ONLY how it appears on screen. Pantone's swatch libraries are based on the measured LAB values of an actual swatch of Pantone Black ink PRINTED on either a coated sheet or an uncoated sheet. e.g. your Pantone Black printed on an uncoated sheet will be a bit more washed out/greyer than the how it would appear on a coated sheet. (Actual Pantone Inks do NOT have different versions for C or U: there's only one Pantone Black.).

If you, however, use the Solid "U" swatches in a process job, your Colour Management settings will convert the LAB value in the Black U swatch and convert it to cmyk (based on your Color Settings) to visually match the printed, slightly washed out/greyish chip, and it will uses a 4 cmyk inks to do it. Even if you use the coated "Black C" swatch, it's measured LAB values are still slightly lighter than a dark Black than you'd expect.

Converely, Pantone created the Color Bridge Libraries, where they have created a set of swatches what would be the best process match in a typical cmyk printing workflow. The swatches for those also have a Coated and Uncoated version, If you chose pantone Black out of either of those, you'll notice both specifiy 0C 0M 0Y 100K