I would assume that this happens because of a difference in the style definitions between the two documents. Style definitions in turn may be based on themes so I would be guessing and say that you need to ensure both documents have the same theme applied and that the style definitions need to match if you want the content pasted between docs to retain the same look.

Editing styles is hard to find in more recent versions of Visio - I add the Define Styles... button to my Quick Access Toolbar for this purpose.

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Andrew Lockton
Chrysalis Design, Melbourne Australia