I'm not sure of the voter registration figures, but in the early 1950s -- when this film was made -- Maine had the reputation as the home of "rock-ribbed Republicans" (an allusion to Maine's rocky coastline) who dependably supported GOP presidential candidates in every election. As a matter of fact, between 1860 and 1960, Maine gave a majority of its votes to GOP candidates in every presidential election (although Democrat Woodrow Wilson won a plurality in 1912 when the GOP vote was split between the regular Republican candidate, William Howard Taft, and the Progressive Republican Theodore Roosevelt). It wasn't until 1964, when Lyndon Johnson swamped GOP candidate Barry Goldwater, that Maine voted for a Democratic president. In the past five presidential elections, Maine has supported the Democratic candidate, so it's no longer the Republican "bellweather" state that is was six decades ago. But in 1953, there would have been no shortage of elephant-lovers in Maine, and even today the GOP is strong in state and local government (governor a Democrat but both U.S. senators GOP).
Since I live in Maine, I decided to put my 2 cents in. The very rural areas of Maine (basically the northern and western parts of the state) are still pretty much exclusively Republican. The Lewiston/Auburn area, the Portland area, and pretty much all of southern Maine tend to be heavily Democratic. The rest of Maine is about half and half. There are also a lot of Mainers who pride themselves on being "independent", but in reality most of them are actually more conservative than liberal.
I believe the OP confused Loco's confusion about what a "lodge" was, not about a Republican rally. So the question isn't based on anything said in the film.
The two comments about Rory Calhoun's father getting drunk at a Republican rally, and his reply to Fred Clark's suggestion that he become a Democrat, were based on the fact that at that time Maine was considered virtually a one-party Republican state. In fact, before the Civil War Maine had been one-party Democratic -- the main reason it wanted to separate from Massachusetts and become its own state in 1820 was that the people there disliked the Federalists who had dominated Massachusetts. But its politics swung in the other direction when the Republican Party was formed in the 1850s, and few Democrats won election during the following century.
Actually, in 1953 Maine was about to end its domination by one party. Democrat Ed Muskie was elected Governor in 1954, and since then the state has had a healthy two-party system, with a predilection for electing some independents as well. Today it's much more Democratic than Republican but people of both parties, and often third parties, regularly win.
Like many of the references in this movie, this political joke is long outmoded.