Tissue compartments do not correspond with any whole organs, they are a theoretical concept to group together tissue that have similar half times for ongassing and offgassing. And lungs would be a particularly confusing organ to use as an example. Do you mean the alveolar space, the vascular space or the supporting soft tissue? The lung certainly aren't a single tissue compartment. It also makes no sense to talk about saturation with respect to gas spaces, since Henry's law refers to a gas-liquid interface.
But for any perfused soft tissue (including the part of the lungs that aren't gas spaces or intravascular), if p(tis)=p(insp) then you have a Henry's law equilibrium and that would be the conventional description of saturation, where there is no gradient for that given gas. Assuming that your p(insp) is 0.79 ATA for N2 and essentially 0 ATA for He right now, I would think that your perfused tissues would have the same partial pressures of those two gases, right? p(tis)=p(insp). So why wouldn't you consider them saturated? Saturated just means that no ongassing or offgassing is happening?
Only in deco speak. Which is why it is confusing. Really my lungs/blood/bones are in equilibrium with p(insp) (ignoring h20, co2 etc). I can go up hills and no gas will come out of solution. If I were genuinely saturated bubbles would form.
Consider a damp sponge on a humid day. It reaches equilibrium by getting damper or less damp. When at equilibrium can you pour water on it and expect it to absorb it? Was it saturated?