It was standard in American mixological tradition to make mixed drinks of one 2 fl-oz. jigger of total liquor (whether from one bottle or more). This was true until prohibition, and remained mostly true even for some time afterwards.
The traditional cocktail goblet was designed to hold a cocktail – being a 2 fl-oz. jigger of total liquor with a slight amount of bitters and sugar, and the amount of water that would be added while stirring or shaking with ice. Though the later standard capacity for the cocktail goblet would become 4½ fl-oz., pre-prohibition cocktail goblets often held only 3 fl-oz.
Sour goblets held more than cocktail goblets because sours were made with the same 2 fl-oz. jigger of total liquor, but with a greater volume of other ingredients added to it than in the case of cocktails.
Though it is elementally a sour, the Sidecar was called a cocktail. Giving it the superficiality of a cocktail meant serving it in a cocktail goblet. By the 1920’s it seems the image of the cocktail had become more important than its classical definition – or the standard of making mixed drinks from a jigger (2 fl-oz.) of total liquor. Note that in the above recipe, the amount for each of the ingredients is ⅙ of a gill. A gill is 4 fl-oz., or ½ cup. That makes ⅙ of a gill equivalent to ⅔ fl-oz., or ⅓ jigger.
I find it probable that the author of the above recipe used substandard amounts for the liquid ingredients so that the total volume of the pre-shaken drink would be 2 fl-oz. Even though the recipe only contains 1⅓ fl-oz. of liquor, the 2 fl-oz. total liquid volume of the recipe would fit the cocktail goblet, even after shaking with ice.
It is interesting to consider that if, before the above book were written, the original creator of the Sidecar used a full jigger of total liquor in it (as per standard practice at the time of the drink’s birth), he might well have made the drink with the same amounts as are common today: 1 fl-oz. of the brandy, 1 fl-oz. of the triple-sec Curaçao liqueur (that’s what Cointreau is), and 1 fl-oz. of the lemon juice. That would mean that after shaking the drink with ice, it would have filled the then-standard-sized cocktail goblet and left some over to be served on the side of the main drink in a small tumbler – hence, perhaps, the ‘sidecar.’