You can activate “The League of Uniform Nomenclature” and Special Summon only 1 copy of the selected monster from your Deck, but the opponent can check your Deck to verify that you do not have a second copy. You can activate “The League of Uniform Nomenclature” targeting a monster like “Right Arm of the Forbidden One” even though it is impossible to have more copies in your Deck (and your opponent can check your Deck to verify).
A Normal Monster on the field that has its level lowered to 2 or less by an effect (like “A Legendary Ocean” or “Demotion”) is a proper target for “The League of Uniform Nomenclature”. So you can target “Archfiend Soldier” equipped with “Demotion” and Special Summon 2 copies of “Archfiend Soldier” from your Deck.
If you target a “Harpie Lady” on the field equipped with “Demotion”, then you can Special Summon monsters that are treated as having the same name, like “Cyber Harpie Lady” or “Harpie Lady 3”.