As you imply, anti-magic affects "spells and magical effects". Spells are obvious. Magical effect means an effect created/sustained by magic - either a spell or magic item.
D&D 5e says what it means and means what it says: unless an ability is called out as magical or a spell then it is neither.
The paladin's aura is not identified as magical so it isn't and is unaffected by an antimagic zone. Neither is their ability to lay on hands, however, a Potion of Healing is because it is a "magical red fluid" or a Cure Wounds spell is because its a spell.
Indeed, most class features are not magical in any way. Exceptions include:
- All classes' spellcasting abilities (duh)
- Barbarian's Totem effects: "... your totem spirit fills you
with supernatural might, adding magical fuel to your
- Cleric's and Paladin's Channel Divinity: "... using that energy to fuel
- Druid's Wild Shape: "... to
magically assume the shape ...". Its debatable if this would cause you to change back or if an antimagic field would only prevent you from "assum[ing] the shape"
- Druid's and Monk's Timeless Body: "... the primal magic that you wield
causes you to age more slowly." You'd have to spend a long time in an antimagic zone for this to be noticeable though.
- Fighter's Weapon Bond: "... that creates a magical
- Monk's Ki: "... a magical energy that most
monastic traditions call ki."
- Paladin's Divine Health: "... the divine magic flowing through you
makes you immune to disease."
- Paladin's Aura of Warding: "... ancient magic lies so heavily
upon you that it forms an eldritch ward."
- Sorcerer's Draconic Resistance: "As magic flows through your body, it causes physical
traits of your dragon ancestors to emerge." Like Wild Shape its debatable that you would lose these in an antimagic field as it can be read that magic creates them but doesn't sustain them.
- Sorcerer's Bend Fate: "... twist fate
using your wild magic."
- Warlock's Eldrich Invocations: "... imbue you with an abiding magical ability."
- Warlock's Entropic Ward: "... magically ward yourself ...".
- Wizard's Arcane Ward: "... weave magic around
- Wizard's Hypnotic Gaze: "... magically enthral ..."
- Wizard's Command Undead: "... use magic to bring
undead under your control ..."
- Wizard's Transmuter's Stone: "... creating a
transmuter’s stone that stores transmutation magic."