Each text box is anchored to a particular paragraph mark. If you copy the
paragraph mark that "owns" the text box, the the text box will be copied, as
well.
To copy just the formatting inside the text box, you need to click inside
the text box, press Ctrl+A/Ctrl+C, and then press Ctrl+V where you want the
text to go.
If you're copying a larger area that includes one or more text
boxes--including the paragraphs where the text boxes reside, then the boxes
will come along for the ride.
Word's pasting does not possess the kind of granularity/control you're
looking for, I suspect. You would have to substitute the granularity
yourself either manually or by scripting a macro. Off hand, I'd think it
would take a pretty smart macro indeed to do exactly what you
want--particularly if the text boxes are not in line with text--since it
would need to figure out exactly where to place the text.
An intermediate solution would be to copy the material into the clipboard,
paste it into Wordpad (which will strip out the text boxes but not the
formatting), copy that into the clipboard, and finally, paste into Word.