You would need to create separate MultiMesh objects for each distinct fish mesh.
MultiMesh depends on using a single mesh.
You can modulate the color of a given MultiMesh mesh instance without impacting other meshes of that object. That documentation you linked outlines near the bottom (the
INSTANCE_CUSTOM section) how one would go about doing it.
Coloring in GDScript:
for i in range($School.multimesh.instance_count):
$School.multimesh.set_instance_custom_data(i, Color(randf(), randf(), randf(), randf()));
MultiMesh custom data is just a 4 component vector (perfect for color+another dimension)
If you did find more fish meshes and want to include them, you should be able to save the motion shader from your first try as a file external to the scene (this is generally recommended, as Godot's scene-specific resources aren't perfect). Then it's just a matter of applying that shader to the new MultiMesh objects with the new, different meshes.
EDIT: forgot to mention the custom instance data is available in the shader as the variable INSTANCE_CUSTOM