This error suggests that either the package is incorrect, or – which is extremely more likely – that it uses the syntax of a newer dpkg version that the older version in trusty doesn’t understand.
So, the workaround is indeed to use Xenial.
Download .NET (Linux, macOS, and Windows), does claim Trusty support but lists a different set of steps.
https://github.com/travis-ci/travis-build/blob/12634169f0b886123b2e219f05f07cb38decd0d5/lib/travis/build/script/csharp.rb doesn’t follow these steps but invents its custom ones that make their own assumptions about URLs and such (how much they are equivalent is up for debate; the important thing is, MS doesn’t promise that any other steps would work).
So this looks like Travis’ issue; could possibly be MS’ issue, too, if the package fetched with correct steps also couldn’t be parsed.