>>10485631
>>10485637
First, it's mostly a show of service from the restaurant. As in "We're not skimping on the alcohol you paid for, you're getting a full glass and beyond"
Second, a portion of alcohol is considered, by law, to be 1合, approx. 0.18L (6oz)
Since most glasses aren't made to contain that quantity (especially with ice), the bamboo receptacle is used to house the excess while still getting what you paid for.
Cold sake also tends to heat up to room temperature rapidly.
The still fridge-cold excess is usually drunk in one gulp at the start, though some like to leave it in for the end so that it can act as a buffer, keeping the sake inside the glass (which heats up faster since you keep pulling it out to drink) cold for longer.