What is the significance of Severus Snape's Petronus being a doe?

I just re-watched the Harry Potter series and I am slightly mystified by this. What does Snape's petronus mean in relation to Lily's (a doe) and Harry's (a stag). For some reason, I can't get it out of my head that Harry is Severus's son, but I know that there are a ton of things that imply the contrary. What are your thoughts?