Hi, I guess what happens is normal and expected, but there is another way to do what you're looking for.
There are 2 ways to select and copy things in a RTE, 2 kinds of focus that the Webflow Designer handles differently.
If you slelect a sub element of an RTE by selecting it's designers' boundaries (with a blue outline) you're selecting the entire RTE, actually. So if you copy paste this, the entire RTE get copied and pasted.
Above: wrong way to select an RTE sub element for editing. This level of selection is meant to handle sub classes for sub content/styling of the sub elements.
To handle content in a RTE you need to be one level down. An RTE content is only a following of strings, so you got to be at the characters level to edit it. Select text like you would do in a word processing program, then copy it and go elsewhere to paste it.
Above: right way to select an RTE sub content for editing. This level of selection is meant to edit the characters strings and to give ensembles predetermined styles such as titles, basic formatting and links.
So I guess that's what you were missing. Before being harsh on the procedure, I'd like to add that RTE and inline RTE is a very hard thing to handle. Close to 20 years now that RTE blocks exist in the CMS world and it has rarely been so well handled than in Webflow IMHO.