"engines" episode explains and gives illustrations of meaning of the word.
http://www.uh.edu/engines/epi1739.htm

from the above link - Thanks, Bill! -
So let us all be merry o'er a bicker of good ale

Does this use of "bicker" have any relation to our modern usage, meaning to argue crossly about petty disagreements? Is that what we do after too much ale? How big is a bicker, anyway?

We have "Bicker (n.) A small wooden vessel made of staves and hoops, like a tub," mentioned in only two out of fourteen online dictionaries. Nobody wants to take a stab as to how much it contains. Do we think it's a longingly big, big drink? or just a modest one?

I couldn't find any estimates of volume. Probably about a pint. And "bicker " to squabble
may be Celtic: [OE. bikeren, perh. fr. Celtic; cf. W. bicra to fight, bicker

If the ale is good what matter the size of the vessel?

OED says merely: 'A bowl or dish for containing liquor, properly, one made of wood.' It goes on to expand slightly on the definition, mentioning "any material" and "made of staves", but I think that there is no specific quantity involved. Empty one, start on the next, I says.

BTW, I make the song Scottish, not Irish and in the verse:

There's soldiers and there's sailors and glaziers and all
There's doctors and there's ministers and them that live by law
And our friends in Sooth America, though them we never saw
But we can they wear the work of the weavers

That should be ken, not can.

It is always strange to me that in English you use "glass" for the material (we say "vetro" in this case) and also for the container of liquid: we call this BICCHIERE.

Emanuela's introduction of bicchiere into the mix drove me straight back to my OED from whence I discovered that bicker, in the drinking vessel context, is a Scots variant of beaker. This led me to my AHD, which told me that beaker is probably from Vulgar Latin bicarium (unattested) via Old Norse bikarr. Ultimately it is possibly from Greek bikos, drinking jar, possibly from Egyptian bik(with a dot under the k), oil vessel. Presumably, this unattested bicarium would be the source of the Italian bicchiere.

Dear Faldage: Isn't it possible that drinking vessels of glass were for many years the only
thing the English coulld make of glass, and so superior and expensive as to rate a separate
word? (Imagine how vile tasting a porous wooden drinking vessel would get.)

The word glass, as applied to a container made of glass, seems to have originally include bottles and jars. The drinking vessel sense seems to have taken over as the sole proprietor of the vessel meaning.

1.Johann Wolfgang von Goethe, König in Thule - Johann Wolfgang von Goethe Der König
in Thule. Es war ein König in Thule Gar treu bis an das Grab, Dem sterbend seine Buhle Einen
goldnen Becher gab. ...