In print documents, the swatch [Black] in InDesign is just plain CMYK black: CMYK(0, 0, 0, 100), so you can just use that in Illustrator.
The special thing about [Black] is that InDesign offers the possiblity to turn on Preferences > Appearance of Black > Overprint [Black] Swatch at 100%. Useful for black text and black lines which (in most cases) should overprint. Illustrator doesn't have that setting because it doesn't have the [Black] swatch. You must manually make sure to overprint black when needed.
Under Preferences > Appearance of Black make sure to set both InDesign and Illustrator to always Display All Blacks Accurately and Output All Blacks Accurately so you have full control over what is going on.
You can use the Separations Preview panel to check which inks your current document is using.