To what microscopic quantity does Gibbs energy correspond to?
As others have said, the Gibbs function (or Gibbs energy) makes sense at a macroscopic, thermodynamic level (instead of a microscopic level), but if you need to picture it, you can elaborate what it means from the deconstruction of its mathematical definition.

Internal energy

Let's start with internal energy, U. Internal energy is the mechanical energy (potential + kinetic) of all the particles in the system. Take into account that it isn't reducible to the energy of separate particles - these particles are interacting (for chemical systems, through intermolecular forces) and the potential and kinetic energy are intrinsically related (though quantum mechanics). The only energy that you aren't considering is the kinetic energy of the system as a whole - if your chemical system is moving through space or rotating around its centre of mass, that energy isn't relevant here (it's also trivially easy to calculate via Newtonian mechanics) - and potential energies that arise from external factors that affect the system uniformly - so if your whole system is subject to gravity, that potential energy isn't computed into the internal energy (also trivially easy to calculate via classical physics). In other words, the internal energy discounts anything that you could calculate if you picture your system as a point mass (as you do in high school mechanics); it deals with whatever is happening inside that point mass.

This mechanical energy is conserved, but the balance between its components will depend on the exact positions of all atoms; there would be constant transferences of energy from kinetic to potential, from inter- to intramolecular interactions, etc. It's easier if we start picturing the system in statistical terms - as a huge amount of molecules, moving in random directions at random speeds (following known statistical distributions), of which thermodynamic functions (such as Gibbs') capture an average.

So, in the internal energy you are taking into account: 1) the kinetic energy of molecules in your system, 2) the potential energy of intermolecular interactions, 3) the kinetic energy of intramolecular movements (rotations, vibrations), 4) intramolecular potentials (arising from the electronic structure of molecules).

Enthalpy

Since we are considering the movement of molecules relative to the centre of mass of the system, we know that the total movement of the system is zero - the centre of mass remains fixed. If you add all the momenta of the molecules, you get a zero.

However, there's a possible collective movement that, while averaging to zero, implies a macroscopic physical change: movements towards the system or away from it. It is possible, if you add all the momenta, to get a total momentum of zero and, at the same time, that more molecules are moving away from the centre of mass than towards it - if you picture it in macroscopic terms, that means that the system as a whole remains at the same position, but is expanding (think of a balloon). And, likewise, if more molecules move towards the centre than away from it, the system is immobile but contracting. Both these collective movements can, potentially, change the volume of the system - they are a microscopic interpretation of pressure.

Note that, ff the system actually changes volume, the intermolecular potential energy is affected - if the system expands, its molecules are further apart and its interactions are weaker; for a contracting system, the opposite is true. The energy involved in that process is purely a mechanical work - captured in the term $pV$. When we work with solids or liquids, changes in volume are usually negligible; when we work with gasses, we usually store them in rigid containers - so changes in volume are also usually negligible. Therefore, it's convenient for chemists to define a function that discounts this mechanical work from the internal energy - that's exactly what enthalpy, H, is.

So, in the enthalpy you are taking into account: 1) molecular movement (except expansion and contraction), 2) intermolecular potentials, 3) intramolecular movement, 4) intramolecular potentials.

Entropy

The molecular interpretation of entropy, S, is always contentious (disregard any explanation that includes the words "chaos" or "disorder"), but we can discuss it simplistically. Entropy is more complex to visualise because it's an intrinsically statistical magnitude, and it ultimately relates to how energy is transferred between the different forms (inter- and intramolecular, kinetic and potential). What entropy implies is that when more than one configuration produces equivalent states, that makes the state more likely - effectively stabilising it, lowering its energy.

If you took microscopic photos of your system at different points, you'd get random positions and states for its molecules - but some situations would be more likely, from a purely random viewpoint, than others. If you have a two-component system, it's very very unlikely that their molecules will be spontaneously segregated in different parts of the system - if you place molecules randomly, it's far more likely that they will be well mixed throughout the system, so, more entropy for the latter. Molecules with many internal movements (like large molecules with many vibrations) also produce more equivalent states - you could get many photos in which the only difference is what particular bond vibrates more strongly, while that's impossible for, say, diatomic compounds. An ideal crystal at T=0K, where atoms are fixed exactly in periodic positions, without defects, and without vibration because they don't have any kinetic energy, has only one way to be arranged - so it has minimum entropy.

The ability to populate these configurations is closely tied to temperature. Temperature provides the energy that produces these movements and allows the system to explore more configurations - through Boltzmann's distribution. Since entropy indicates how many configurations there are for a given state and temperature provides the energy to do so, the "entropic stabilisation" term depends on temperature, and, conversely, systems with higher entropy are more sensitive to entropic stabilisation. That's what the term $TS$ measures.

This is important because that "entropic stabilisation" means that a fraction of the energy of the system (precisely $TS$) is tied up in providing these alternative configurations, and cannot be extracted directly from the system. Unless chemical reactions or other structural changes take place, that energy is inaccessible.

Gibbs function

We can then describe a function that computes 1) the internal mechanical energy of the system, 2) discounts the energy involved in expansion/compression work, and 3) discounts the energy that is inaccessible due to entropy. This is what the Gibbs function, G, means at a microscopic level. Note that this doesn't make for a clear-cut distinction between what microscopic forms of energy are included in the Gibbs function - it isn't as simple as "kinetic out, potential in", because some things (mainly entropy) are statistically defined and only make sense for a collective of (interacting) particles.