To my knowledge this was possible in earlier versions, but now, using arcgis 10.4 it doesn't work. As long as the label is on a straight line everything's fine - it's just the curved instances, or even if the label was placed by maplex engine in two lines - the two lines become two distinct text boxes.
Try to use Insert Object.
Or
export your curved label to picture
I'm using ArcMap 10.2 and it's the same behaviour. This old topic talks about that: ArcGIS Labeling to Illustrator - Software Applications - CartoTalk so I think the issue is not new.
When you are labeling line features, especially if you use the Maplex 'street placement', you can get several cases in Illustrator side:
- each letter separated
- each word separated as text on path
- one straight text
This behaviour is very disappointing for me. Ideally, I would get only texts on path, where path follows the whole street line. That would be so easier to adjust manually in Illustrator.