The only thing that can be accurately copied from Illustrator to InDesign are vector graphics (as long they are relatively simple).
Fonts won't copy because InDesign has a more advanced text engine than Illustrator has. They are simply incompatible. Nothing new, it's always been that way (I've been an Illustrator user since Illustrator 88, InDesign since InDesign 1.0 in 1999).
However, you didn't mention what version of each application you're using.
I'm using InDesign CC 2015.2 and Illustrator CC 2015.2.1. What I was able to do was to copy small amounts of text from Illustrator to InDesign using the Type tool. Of course, it came in unformatted as it usually does.
But in Illustrator, I used the Libraries panel and added paragraph styles and character styles from the Illustrator file to my Libraries panel. I returned to InDesign, and applied the same paragraph and character styles from the CC Libraries panel (which as access to the same assets) to the unformatted InDesign file.
That would be workaround. It's a bit tedious, but it may be the best you can do.
These jobs are CS6 for InDesign and Illustrator, both on Mac. Tried it in CC and I got the same results. Its works for me. I think maybe the client is fibbing about loading the fonts I sent.
Why not embed the fonts in the AI file and place that instead of copy/paste?
Peter's method would be the easiest and would work if you didn't need to actually edit in InDesign.
I was assuming that (as can happen), the job later needed to be edited in InDesign.
Time, speed... imagine opening 26 documents, saving 24 separate pieces of art out of each in 3 different colors each, then placing each in 26 new InDesign docs. All in one day. I'm not an agency that has weeks to do jobs, I get minutes to hours. That is 1872 pieces of artwork. I can do it with cut and paste in less than 1/5th the time (I can cut and paste all 24 pieces of artwork at once, ungroup, and move into position, per doc).
Turns out I am right. The fonts do NOT need to be outlined, or saved as eps. It works fine.
Sorry for the thread
You should never, in any case outline fonts in InDesign nor use EPS.