Q. You advise capitalizing the shared generic term in topographical names (“the Illinois and the Chicago Rivers,” CMOS 8.53). Do you advise the same for other things, such as churches (“the Anglican, Armenian, and Catholic Churches”) and parties (“the Democratic and Republican Parties”)?
A. Yes, Chicago’s rule for rivers, mountains, and the like would normally extend to other types of proper nouns—including the names of political divisions (CMOS 8.51), streets (CMOS 8.56), and buildings and monuments (CMOS 8.57). As with those categories, the rule would apply to churches and parties only when each of the formal names (or sometimes a shorter version thereof) incorporates the generic term, capitalized as part of the name—the Anglican Church, the Democratic Party (in the US), and so forth.
As rules go, however, this one is pretty arbitrary. It took three editions of CMOS to settle on a recommendation for the plural forms of topographical divisions. The 14th edition introduced the current recommendation (which had formerly applied only when the generic term preceded the names: Lakes Erie and Huron)—only to have it reversed for the 15th and then (after an in-house poll and input from readers) reinstated for the 16th. As for churches and parties, these weren’t capitalized even for singular entities until the 14th edition (the 13th listed “Republican party”; its Democratic counterpart was absent from the list). So a preference for lowercase wouldn’t be unreasonable—particularly for churches and parties. Just be consistent.