Green’s Dictionary of Slang

1. (US) a large glass of beer, slightly less than a pint.

1893

Sun (NY) 21 Apr. 7/2: If the hoister is on intimate terms with these schooners [of beer] he familiarly calls them ‘tubs’ or ‘high hats’. Some prefer to call them ‘geesers’.

2. (US) a drink of whisky or strong alcohol.

1902		Times (Shreveport, LA) 12 May 3/5: A ‘geezer o’ skat’ is a drink of whisky.
1914		Jackson & Hellyer Vocab. Criminal Sl.
1931		G. Milburn ‘Convicts’ Jargon’ in AS VI:6 438: geezer, n. A drink of any high-powered booze.
1935		A.J. Pollock Und. Speaks.
1950		Goldin et al. DAUL 77/2: Geezer. [...] 2. A drink of potent liquor.