The splined shaft and bearings inside the frame are considered to be the Bottom Bracket assembly (BB).
The BB splined shaft is steel and unlikely to have been damaged by the softer aluminium crank arms or more specifically the teeth/splines inside those crank arms.
Not fitting (torquing) a crank arm correctly can result in it spinning on the splined shaft and damaging the soft aluminium splines/teeth in the crank arm.
Riding a loose crank will accelerate the damage.
I'd suggest cleaning those splines on the shaft and crank arms. Then inspecting the crank splines for damage.
Given that you suffered some spinning it is likely you have damage inside one or both crank arms.
If the damage is limited to the Non Drive Side (NDS) (the side without the chain ring) then replacing just that crank could be a cheap fix.
If your inspection shows damage on the Drive Side (DS), well... time to pay up.
Whatever you choose to do regarding parts, do
get the correct torque when performing the reassembly.
If I'm correct, it looks like about 30 foot/pounds. You are unlikely to achieve that using a ~6" allen wrench.
All the best
Barry