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Does the potential energy scale in the above mentioned graph measure the Gibbs free energy of the system?

As the system always tries to achieve a minimum of free energy and chemical reactions are taking place can be considered a non-mechanical work (I'm not sure of it, as the system isn't in a state of chemical equilibrium if a reaction is taking place), potential energy should represent Gibbs free energy.

If not, how should I understand the potential energy of a reaction mixture?

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There is a long explanation for that, however in short they are not the same thing. Free energy relates to thermodynamic systems with many particles and is governed by Boltzmann's distribution, a sum of all possible energy configurations, where F is the free energy of the system.

$Z = e^{- \frac{F}{kT}} = \sum_i e^{- \frac{E_i}{kT}}$

Potential energy is non-statistical and well-defined for one particle. Thermodynamic free energy (one type being Gibbs) is a subset of the total potential energy. Even though they may seem similar, they should not be confused and should not be interchanged.

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