Most prominently featured of these is Bacchus - the Roman god of drinking and parties who shows up drunk and riding a unicorn donkey and inspires a bacchanal (a big wild drunken party in which many of the revelers hook up in romantic situations).
Some fun facts about Bacchus (or Dionysus, as the Greeks called him):
1. Bacchus's dad, Zeus, took the form of lightening and scared fetal Bacchus right out of his mother's womb only six months into the pregnancy, killing her in the process. After that, Zeus sewed the baby up into his thigh where he spent the remainder of the gestation period. Apparently that works**.
2. Since Zeus's wife, Hera, was not Bacchus's mother, Zeus had to keep the child hidden from her jealous rage. To do this, Bacchus spent some of his childhood being raised as a girl, and some of it being raised as a ram. I think this goes a long way to explain the constant state of drunkenness in his adulthood.
Henry Wadsworth Longfellow wrote of Bacchus in his poem, Drinking Song. In it, you'll see some echoes of moments in Fantasia. A few choice stanzas:
Fauns with youthful Bacchus follow;
Ivy crowns that brow supernal
As the forehead of Apollo,
And possessing youth eternal.
Round about him, fair Bacchantes,
Bearing cymbals, flutes, and thyrses,
Wild from Naxian groves, or Zante's
Vineyards, sing delirious verses.
These are ancient ethnic revels,
Of a faith long since forsaken;
Now the Satyrs, changed to devils,
Frighten mortals wine-o'ertaken.
Sadly, I could find no reference in mythology to Bacchus riding a unicorn donkey ***, but other than that, Fantasia's portrayal of him seems pretty appropriate.
If you're interested in further reading about Bacchus:
Myths about the Roman God Bacchus
Theoi Greek Mythology: Dionysus
Bonus fun fact: Along with many of the gods in Fantasia, Bacchuss (now wine colored) shows up again in Disney's Hercules.
*I'm so sorry. This joke is terrible.
**Probably only for Gods.