In the Netherlands, there existed so-called 'pillarisation' ('verzuiling') from the late 1800 onward, which only began to disappear after World War II. Society was vertically segregated, and the main political and religious groups within society were explicitly viewed as separate 'pillars' of society. Protestants, catholics, social-democrats etc. all had their own newspapers, radio channels (and later TV channels), trade unions, political parties, schools, insurance companies... even banks, hospitals and universities were often aimed at one 'pillar' in particular. Society, needless to say, was extremely segregated (though in a rather voluntary way, that was explicitly not aimed at placing one group above the others - they were all equal pillars, needed to uphold society as a whole).
A similar system existed in Belgium, and has existed in various other countries.
If such a system were to become the fashion in the USA after the Civil War, one can easily imagine the various highly segregated social groups to also have their own militia forces. These would likely evolve from existing Marching Clubs and similar organisations. In fact, these might in some cases largely replace the police. (The idea being that every community/'pillar' organises its own police to take care of its own people.) I don't think the implementation of such a system is very likely in the USA - largely because it will probably end the two-party system and cause all 'pillars' to establish their own parties - but it might just come up as an ATL solution to relieve social and racial tensions post-Civil War. (The age-old "let's just all stick to our own"-approach.)
Furthermore, a single POD - increased social and racial tension, compared to OTL, in the postbellum period - can justify both the stronger position of the Marching Clubs and increased support for pillarisation (which would serve to keep the Marching Clubs going later on, even after tensions die down).