1659 from Scots powny, apparently from Middle French poulenet, ultimately from Late Latin pullanus, from pullus (cognate to English foal). Sense “small serving of alcohol” from 19th century, both for small sizes generally and for a quarter pint specifically, from the small size.Notes and Queries, August 8th, 1896, : “It seems probable the origin is due to the diminutiveness of the glass;”
“The expression ‘a pony of beer’ is often used in South Wales for a small glass containing about the fourth of a pint.”