There is no reason to expect life to survive an asteroid impact. The sheer force would likely be enough to crush delicate cells, and the heat of the impact would likely sterilize any remnants.
Also, there are no fungi living off frozen water at -50 to -70 C or below on Earth, and an asteroid impact doesn't really provide time for evolution to work its magic. Perhaps extremeophiles could be found that would survive, though even that is shaky without some form of liquid water, and they would be extremely unlikely to evolve to something approaching fungi (remeber that extremeophiles are prokaryotes, while fungi are eukaryotes, and as far as we can tell eukaryotes arose from prokaryotes only once in the history of life on Earth, so it was literally a once in 2 billion years event).
Edit to add: Mars' surface is also extremely toxic to essentially all Earth-based life.
Overall, it is very hard to imagine that any form of Earth-based life would have survived the series of events you suggest: an asteroid impact with Earth which propelled a chunk of rock to space, being exposed to space, then burning through the Martian atmosphere, crashing into Mars, then encountering the extreme cold, low pressure, missing oxygen and CO2, high radiation, the toxic soil of Mars, and not just surviving, but growing and thriving from this.