What is the point of the Tankard?

One of the items you can use is a Tankard. It is refillable from bars or from your ship's grog barrel.

After consuming the contents of it, you become severely intoxicated, making it hard to even walk. Are there any other benefits to drinking from the Tankard (melee strength bonus, damage reduction, etc), or is it purely for fun?

There are no real benifits of it. You could technically say you could use the vomit to your advantage by blinding enemies in a PVP fight but other than that it is just a bit of fun to be had on those long voyages across the water.

Tried playing some tunes after drinking, it's very entertaining :)

There are a few achievements that can be gained by drinking:

• I'll Drink to That [5G]
  "Nothing says being social, like crews drunk down the local."
• What Shall We Do With a Drunken Sailor? [10G]
  "A tankard or two may make you blue, have a few more and you will surely spew."
• A Titanic Ensemble [10G]
  "A pirate once said that when you are sinking, playing some music is more helpful than drinking."
• Tactical Chunder [20G]
  "There is an old saying that to win a sword fight, a tactical chunder will ruin their sight."