How can you insert text into an already rotated field at the same angle? Is that possible or is the text always horizontal or vertical and than you have to adjust with the desired angle?

There's no direct way, but you don't necessarily have to measure the angle manually. Two similar free scripts might help: 
1) https://github.com/sky-chaser-high/adobe-illustrator-scripts#rotatetexttomatchpathsegmentanglejs
2) https://github.com/gau/adjust-text-angle-to-segment

Video tutorial: 
 

I am afraid not that I know off

Can you share a sample Illustrator file that contain what you’re calling “rotated fields”?

Also, is the question about single line text entries or multi-line?

You cannot use the Area type tool to fill it using an angle. You must use Point type to rotate it.

​@flavorflave Turn on View > Smart Guides and you can use the Measure tool (under the Eyedropper) to measure a side of the rectangle. Use the result to rotate the text

If the rectangle is a shape, you can find the rotation in the Transform panel and copy the value.