What do you mean by stats? If you mean the properties of the move itself, like damage, cooldown, etc., then yes, copies of the same move will level up the same way.
If you mean the stat boost modules attached to the move, like crit resist, defence, element bonus, then no; which one of those gets upgraded is completely random each time and is chosen every level.
Lets's say I have 2 amazing Brass Knuckles. Each of them have ATK 3%, DEF3%, HP3%.
At level 1, they are completely identical.
However, every time I level them up, one of those stats is chosen randomly to be increased. If I level them up to max, their stats can diverge wildly.
One might be ATK 30% DEF3% HP15%.
The other might be ATK 15% DEF15% HP15%.