If two quantities have the same dimensions, do they represent the same physical content then?
Not really. Energy and torque, for example, both have the same units but are very different physical quantities. Energy is conserved, torque is not. Energy is a scalar, torque js a vector. They are very separate concepts despite having the same units. The same is true of angular momentum and action. Angular momentum is a conserved quantity, whereas action is a functional of the path a system takes that is always minimised. Having the same units doesn't show similarity between quantities in every case.