Green’s Dictionary of Slang

(US) a short-lived, unstable bank; thus the money it issues.

1838		N.Y. Advertiser and Express 21 Apr. 1/4: Aptly enough are these banks called ‘Wild Cat’ and ‘Red Dog’.
1840		Morning Herald (N.Y.) 31 Mar. 2/2: Note Holder. [to Banker] Well then, give me a red dog for it — for any kind of note is better than that.
1859		Bartlett Dict. Americanisms (2nd edn) 357: The community stigmatized them as red dogs.
1886		Mag. Western Hist. III 202/2: The banks toppled to the earth [...]. They were known universally under the name of ‘wild-cats.’ The most worthless were styled ‘red-dog’ [DA].