These financial senses predate the political ones (from the OED):
1761 H. WALPOLE Lett. H. Mann 28 Dec. (1843) I. 60 Do you know what a Bull, and a Bear, and a Lame Duck are? 1771 GARRICK Prol. to Foote’s Maid of B., Change-Alley bankrupts waddle out lame ducks!
There is this nautical sense from about the time the political term came into vogue:
1876 C. CHAPMAN First Ten Yrs. Sailor’s Life at Sea x. 411 A lame duck on the sea means a ship which has been more or less damaged while crossing the perilous ocean.
And there is this, the first cite in a political sense, which doesn’t sound as if it refers specifically to politicians who have lost an election, although I’d have to find the source and read the context to be sure:
1863 Congress. Globe 14 Jan. 307/1 In no event..could it [sc. the Court of Claims] be justly obnoxious to the charge of being a receptacle of ‘lame ducks’ or broken down politicians.
So the primary sense has always been “damaged,” with a secondary, more specialized sense of “serving out the remainder of a term after having lost an election.”