It's a humorous way of describing a shower that is:
* Short and to the point: You get in, get wet with cold water, and get out quickly.
* Not particularly luxurious: There's no time for warm water, bubbles, or pampering.
The term likely comes from the stereotype of Irish people being tough and resilient, able to handle a quick blast of cold water without complaint.
Of course, it's just a playful expression, and not everyone who takes a cold shower is Irish, or enjoys it!