An old expression dating back at least to British National Service (1950's) and probably a lot earlier. Possibly for as long as there have been British Armies that supplied blankets.
Was also commonly used in former British Territories and Dominions (Australia etc.) which now form the Commonwealth.
To attach to something very securely.
Also to hang around something or someone very persistently.
"This glue will stick that on the wall like shit to an Army blanket."
"He's stuck to his boss like shit to an Army blanket."