Google can't even tell that the search is coming from DuckDuckGo, as DDG sends this when redirecting to a Google-owned service:<p><pre><code> <meta name='referrer' content='never'>
</code></pre>
Interestingly, it doesn't to other networks. For example, Wikipedia, Reddit, and Twitter redirects use this:<p><pre><code> <meta name='referrer' content='origin'>
</code></pre>
So they can tell it's from DDG, but not the exact URL (not that it matters because they get the query anyways)
Are you asking if Google accounts for the bangs? You probably won't get a straight answer, but it's certainly possible. I believe Google keeps track of the order of your searches, so they might take that into account if you remove the bang and click through to what you were looking for.