Counters don't have the enchantment card type. They don't even have a card type. They aren't even objects.
300.1. The card types are artifact, creature, enchantment, instant, land, phenomenon, plane, planeswalker, scheme, sorcery, tribal, and vanguard.
109.3. An object’s characteristics are name, mana cost, color, color indicator, card type, subtype, supertype, expansion symbol, rules text, abilities, power, toughness, loyalty, hand modifier, and life modifier. Objects can have some or all of these characteristics. Any other information about an object isn’t a characteristic. For example, characteristics don’t include whether a permanent is tapped, a spell’s target, an object’s owner or controller, what an Aura enchants, and so on.
109.1. An object is an ability on the stack, a card, a copy of a card, a token, a spell, a permanent, or an emblem.
121.1. A counter is a marker placed on an object or player that modifies its characteristics and/or interacts with a rule, ability, or effect. Counters are not objects and have no characteristics. Notably, a counter is not a token, and a token is not a counter. Counters with the same name or description are interchangeable.