COVID-19 Outbreak (Update: More than 2.9M cases and 132,313 deaths in US)

So many of these tests I presume are only able to give you a positive or negative result on Covid-19.

Isn’t it possible that a different virus could be to blame for a certain percentage of the flu-related deaths?

I’m assuming you could have enough Covid19 in your system to register a positive sample while also having another virus in you.
So, first, the deaths from Covid-19 are, if anything, understated. If you take a look at this Financial Times analysis from yesterday, comparing current mortality statistics against normal levels, you can quickly see two things: Covid-19 is causing large spikes in mortality across the world, and the number of excess deaths compared to normal levels is still substantially higher than that currently attributed to Covid-19.

As the article discusses, there will be multiple factors to that, but the largest is likely to be that many deaths that are being caused by Covid-19 are not being attributed to Covid-19 due to a lack of specific diagnosis, typically due to a lack of testing (other factors include mortality from other causes increasing where healthcare systems have been overwhelmed due to Covid-19 and become unable to provide timely treatment for other conditions; at the same time, mortality from some causes (e.g. traffic accidents) will likely to have been reduced due to lockdowns).

As for your question, you can have Covid-19 and influenza at the same time, but it's relatively unlikely. The number of people who do will be a small subset of all those with Covid-19, and of those with both who die, the hypothetical number of them who die entirely from influenza and not at all from Covid-19 would be even smaller, even if it were possible to accurately determine. It's not going to be significant outside highly specific contexts.