The search-term you are looking for is Inverse Kinematics. This is a technique you can use to modify character animations to take the positions of other objects in the game world into account. The basic application of this technique is to have characters walk properly on uneven ground or accurately reach for objects in the game world. Having a character climb another character which is animated independently from it is a very advanced use of IK, so you might want to familiarize yourself with the basic applications first.
Now how to do climbing with IK?
Define some vertices of your giant model as "grip vertices". These are points on the giant-model you want your player-model to be able to grip to.
Now when you have the player climb the giant, there will be moments in the climb animation where one of the limbs of the player needs to move from its old grip-vertex towards a new grip-vertex. Do that by picking the vertex closest to the preferred end position of that animation phase and use it as the new end effector for the limb.
By the way, if you are interested in the animations of Shadow of the Colossus specifically, then you might find the Extra Frames video about its IK system very interesting. It also starts with a general introduction of IK and then shows how they are used in SotC.