Green’s Dictionary of Slang

1. a fur, a tippet, a fur coat.

1984

Partridge DSUE (8th edn) 745/2: –1932.

2. a cat.

1927		W.E. Collinson Contemp. Eng. 26: As school-boys [...] Tike for dog, moke for donkey..mog for cat were quite usual.
1953		J. Franklyn Cockney 267: I don’t mind you borrowing our old mog, he’s a grand mouser, he is.

3. a monkey.

1941

V. Davis Phenomena in Crime 43: A tiny monkey with exceptionally long arms [...] ‘What do you eat, Mog?’.