The "Search with <search engine>" context menu action will search with whatever browser you are currently in. When I'm in Orion, selecting text on a webpage and pressing search opens the text in Orion. When in Firefox and using the context menu action, the search is performed in Firefox.
TL;DR the context menu action respects the current browser you are in, and takes priority over the user's browser. I don't think this is something Orion could manage. Even when using Safari as default, the context menu action is browser specific.