Legacy format

The warning box you describe is a default message that pops up whenever saving an Illustrator file in an older version of Illustrator. It also happens by default when you open an older version Illustrator file in Illustrator 25.2 and then hit the Save button. Illustrator will, by default, respect that older file version rather than automatically save in the latest version. Most rival graphics applications will automatically save in the latest file version automatically when the "save" button or command is clicked. To get more control over the process click "Save As."

That default warning box is nothing to be worried about it UNLESS you are using features in that layout that are unique to newer versions of Illustrator and are saving down to a version of Illustrator in which those features are not supported. The new repeats function is one example. OpenType Variable Font support goes back only so many versions. The same goes for free-form gradients. The list goes on and on. If you are only using the latest versions of Illustrator in conjunction with After Effects there shouldn't be any problem.

Exactly how are you saving it? Which options?

What's in the file?

Did the document start at your school?

In that case your school probably uses an older Illustrator version.

If you want to continue working on that Illustrator file at school, leave it as it is when saving, you are saving it in the older version (legacy) format from your school.

If you want to use it in the latest version of Illustrator and After Effects, do a save as version Illustrator 2020. This cannot be opened by Illustrator CC.