I was reading Dan Brown and found out that Dionysus's birthday (I assume 2nd birth from Zeus's calf) is the same as Jesus.
What similarities do they share other than this?
The relationship between these Jesus and Dionysus has multiple aspects. Here are the most basic:
Both had a divine father and a mortal mother
Both were sacrificed
Both died and were resurrected
Wine symbolizes the blood of both
In the Christian tradition, Jesus says "for this is My blood of the covenant, which is poured out for many for forgiveness of sins." [Matthew 26:28] In the pagan tradition, each year Dionysus is symbolically torn apart* in the cutting and stamping of grapes, buried in a cave (typically in a cask), and in spring is "reborn in spirit", literally.
Dionysus also famously "returned from the East" after his resurrection. (Some accounts have him returning from the South, but in both cases he was regarded as a foreign god. This is part of the reason Pentheus, "Divine Agony", rejects him.) Jesus "was from the East" from a Greek perspective, certainly foreign, and came to Greece through the Apostle Paul.
From a mythological perspective, these gods are both "liminal", meaning that they move between worlds, in this case, life/death or earth/heaven.
Dionysus is also known as "twice born" because he was first carried by his mother, Semele, and then removed on her death and borne in Zeus' thigh until his second birth. Jesus is often regarded as having been "born again" and the term is still used today in many Christian sects to describe adherents.
*The sacrifice of Dionysus is sometimes referred to as the sparagmos where he is torn apart and eaten by Titans. This event in commemorated in the Bacchae, which has a religious function--Greek dramas were presented at the Festival of Dionysus.
You may find this verse from TS Eliot interesting:
"The dripping blood our only drink,
The bloody flesh our only food:
In spite of which we like to think
That we are sound, substantial flesh and blood—
Again, in spite of that, we call this Friday good."