You are correct that Union Infantry privates were not issued swords or sabres. Standard issue arms for Infantry privates consisted of a Rifled Musket with bayonet.

Sabres were generally issued to privates and other ranks in the cavalry as well as infantry officers and NCOs, artillerymen etc. Not all the soldiers who had the position to carry a sabre did so, revolvers were often preferred as a side arm if they could be obtained.

The current sabre is an model 1860 light cavalry sabre, so it would have been issued to cavalrymen, but was also popular among infantry and staff officers.

Unless family stories exist, or you can find your ancestor by name in one of the histories of the 6th Kentucky, you probably won't find out exactly why he had each sword.

One thing that happened in Civil War portraiture fairly regularly was prop weapons in the photographer's studio. You can frequently see soldiers in civil war portraits posing with outdated non-regulation weapons, likely because they were props. A civil war photographic portrait was a special occasion in which dressing up a bit more fancy than normal was quite common. (Occasionally soldiers went the other way and posed with Bowie knives to look badass or no weapon) It was pretty much up to the individual how you had your portrait taken.

Discipline in the western theatre of the Civil War could be quite lax. Company and regimental Officers were most often elected and expected to deal with their men in civilian life after the war, since regiments were from one particular area. Looting in the western theatre was common and often officially sanctioned. A model 1860 sabre is only 2-3 pounds, so it makes a perfectly reasonable souvenir. They were also easily purchased post-war.

Basically, your ancestor posed with a sword and acquired a sword as an infantry private, more than likely just because he liked swords.