You can add text from your document to Creative Cloud libraries and then share them with other users and across applications. Text assets in a library retain the character and paragraph styles and other attributes originally applied to them. You can reuse text assets in libraries across Illustrator or InDesign documents. For more information, see Creative Cloud libraries - collaborate, sync, and share assets.
When you use a text asset from Creative Cloud libraries, the rendering and composing is done by the application in which you are using it. For example, if you add a text asset in Creative Cloud libraries from InDesign and use the same in Illustrator, the composition of the text is done by Illustrator. Therefore, you may see some composition differences. What you get is live text with the exact same features applied to it as were in InDesign. If some InDesign feature is missing in Illustrator, it would be dropped or not rendered in Illustrator.
You can add a shared InDesign text asset as a graphic asset in Illustrator to get the same visual appearance. To add a text asset as a graphic, press Ctrl while adding the asset.
Here are some examples where the features and attributes of a text asset created in InDesign are rendered differently in InDesign and Illustrator when the asset is used.
InDesign has more Underline options as compared to Illustrator. The options that are not available in Illustrator are lost while using the text asset in Illustrator.
Bullets and numbering are not supported in Illustrator and therefore these attributes are lost while using the text asset in Illustrator.
Tables are not supported in Illustrator and therefore this feature is lost while using the text asset in Illustrator.
Footnotes are not supported in Illustrator and therefore this feature is lost while using the text asset in Illustrator.