The hand crossbow has to properties, ammo, and light, but you need a free hand to load the crossbow so how is it light if you need both hands.
Loading ≠ making a ranged attack.
The Player's Handbook description of the light weapon property begins
A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
A loaded hand crossbow does not need two hands to fire. For example, see the text of the crossbow expert feat in the Player's Handbook :
When you use the Attack action and attack with a one-handed weapon, you can use a bonus action to attack with a
loaded[since revised] hand crossbow you are holding.
Because the description does not say "you may drop or sheath your one-handed weapon, you can use a bonus action to attack with a loaded hand crossbow you are holding," and because it says "holding", the necessary implication is that you are making the crossbow attack with a single hand.
Sage Advice makes this distinction between firing and the loading aspect of the attack action explicit:
A hand crossbow is a one-handed weapon, so it can, indeed, be used for both attacks, assuming you have a hand free to load the hand crossbow between the two attacks.