This should not happen. When you have a RTE and place a style on it - such as bold then the proper behavior is to bold the text. This appears to be a bug. Please file a support ticket so this can be fixed.
By the way - i am not able to reproduce your issue. I have 2 RTE in my dialog and I can apply a style to text in the 1st one. See --
See this article - notice the 2nd package has a Multifield with a RTE. You can have many RTE in the dialog. This is letting me apply styles to each one.
we had the same issue and found a workaround for it. In case you're having the issue, you can solve it by overlaying the RichText JS. Just do following:
1. Overlay JS "/libs/clientlibs/granite/richtext/js/rte/CUI.RichText.js"
2. Replace its line 331:
In our case that was making the trick.