I think what the author is referring to is that other than beer (and cider) the SI units are used to measure booze in the UK.

You don't even tend to buy beer (or cider) by the pint outside of somewhere "licensed to sell alcohol for consumption on the premises" (ie a bar, pub, some restaurants). You order a pint (or a half, or a third, of very strong beers or to taste test several beers) to drink now, but when buying cans or bottles of beer for consumption later they're labelled in SI units like any other beverage e.g. a 330ml can or a 500ml bottle are perfectly normal things to see.

In a licensed premises the stronger classes of booze are measured in SI units too but with a convenience factor for ordering and, for weird historical reasons, each place of business gets to choose one of two definitions. For example a glass of wine could be 150ml or 175ml. Customers might say "A glass of the house red" but there must be a notice posted defining this as either 150ml or 175ml. Or vodka may be sold with a shot being defined as 25ml or as 35ml (again a posted sign tells you which) so you might say "A double vodka" and that's defined as either 50ml or 70ml.

Though they're not technically required, many British licensed premises use "optics" which are a device that fastens to a liquor bottle, the bottle is then inverted and the optic displays to the customer that there's a fixed measure of product (e.g. 35ml of whisky) ready to dispense, then a single motion dispenses that product into the glass without need for the staff to measure, when the glass is removed the optic refills ready for the next.

Yeah I worked in a 35ml bar and we were supposed to clarify with every single customer who asked for a double that it was going to be 70ml, really tiresome

The UK labeling laws require alcohol is listed in "Units" which are equivilent to 10ml of pure ethanol.