The Quincy 210 was the smallest recip compressor Quincy built. From the photo, the ones on the unit pictured are quite old as they have the rotary vane type lube oil pumps. Those Quincy compressors had full-pressure lubrication to the crankshaft mains and rod end bearings. They use an oil filter screen, no oil filter with a paper element. The later models in that series went to an internal gear pump for the lube oil, and a small spin-on filter. If the vanes wear out, you are on your own to make new ones as Quincy does not support those older compressors.
The unloader system on that compressor is tied into the lube oil system. When the compressor starts up, there is initially no lube oil pressure. The unloader control and springs hold the intake valves open (the head unloader units). Once lube oil pressure builds, the unloader control will release the intake valves and the compressor starts pumping air. On coast-down, the same process happens. The unloader control will be on the side of the crankcase and is tied into the pressure switch.
The 210 compressors are great little compressors. I own one I got used over 40 years ago, ran it until the lube oil pump vanes gave out. Having an even older Worthington compressor, I put that into use for shop air.
The Quincy compressors really are not made to be sped up to increase air flow. A duplex compressor of the type you picture likely came out of a building's HVAC (heating, ventilating, air conditioning) control system. The setup you picture was made for making "control air". The two motors were worked off a separate controller which usually had a "lead-lag" or "alternator" logic to it. This could run one compressor in the "lead" mode, and if it could not keep up, the second compressor came on line. Or, in the "alternator" mode, it alternated which compressor was next to be started when the pressure switches called for air. This was to equalize run times.
A 210 compressor should not need much of a motor. 1 HP would be plenty for what cfm the compressor produces. A 1725 or 1750 rpm motor is what is needed. These are SLOW turning compressors. Nice and quiet. Very long lived if you take care of them. 200 bucks is a steal, even if you have to buy a motor.
The double shaft idea, unfortunately, will not work. This is because the Quincy compressors can only be turned in one direction due to the lube oil pump and internals. Similarly, reading your post, you cannot rotate one of the compressors 180 degrees to run off the same motor from your old compressor. There are directional arrows on the flywheel pulleys of the Quincy compressors.
About the only way to use your old motor would be kind of a "kluge" or "Rube Goldberg" lashup: namely, you'd need to make a jackshaft driven by belts from the motor. This jackshaft would be run in pillow block bearings and would run at 1:1 with the existing motor shaft speed. Each compressor would then be belted off the jackshaft with pulleys the size of the motor pulley that is now used. In effect, you would be making a "mini line shaft". Cumbersome lashup, for sure. it would also start both compressors in parallel with each other. Chances are the existing setup has two separate tappings on the air tank, each with its own compressor check valve. This lets the compressors push air into the tank independent of each other. If you went for the setup I've described, my own gut sense would be to make an "air manifold" out of a chunk of maybe 3" pipe and a couple of butt weld caps. This manifold would provide a bit of dampening so both compressors could discharge into it without having the pulsations from one compressor's discharge bucking against the other. I use "Thred-O-Lets" (forged steel fittings, tapped for pipe threads, ready made to saddle onto pipes that you then weld in place) and butt weld fittings for this sort of thing as well as for other pipe work.
3.7 HP motor sounds a bit odd for a rating. Is it an imported motor ? If it is 3450 or 3600 rpm, as is used on some of the newer consumer-grade compressors, you will need to have a slower turning pulley to drive the Quincy compressor(s). The flywheels on the Quincy 210's are not all that big. I forget the spec'd maximum rpm on a 210 pump, but it may be down around 450-600 rpm at most. The Quincy 210's "perk along" like an old coffee percolator, nice and slow, and not loud.
My other recommendation is to at least do a hydrostatic (water pressure) test on the air tank before you put the unit into service. We've posted quite a bit on this 'board about the dangers in old/unknown air tanks. They are considered as potential bombs. Condensation from the compressed air corrodes and thins the bottom of the tank, and while the tank appears sound, it can be a real bomb waiting to go off. Ideally, gauging the tank with an ultrasonic thickness gauge (UT) followed by the hydro test are what should be done. The hydro test is done with water and the tank is pumped to 1.25 X working pressure. If anything lets go, it is in "slow motion" with no flying shrapnel- as would be the case if a tank containing compressed air were to rupture. Tanks built under ASME (American Society of Mechanical Engineers) unfired pressure vessel code (there will be a "code plate" tack welded to the tank shell) will be built in a National Board shop and have a registration number, date of manufacturer, maximum allowable working pressure, maximum temperature, and thicknesses of the barrel and heads. Tanks are designed with a minimum factor of safety of 5, which is good and bad. A tank can pass hydro test, but be so thin as to be right on the edge of a failure, and the factor of safety has dropped to 1. While the tank passes its hydro test and does not bulge or rupture, all it takes is a little more corrosion internally in service to have an unexpected tank rupture. Chances are the tank will prove out OK, as these sorts of duplex compressors in HVAC systems usually had automatic tank condensate drains on them. Nice easy service compared to being used in a shop or similar.
As I said, it is a steal at the price, and I'd grab it. Even if you wind up buying a smaller air receiver and new motor, you have a spare compressor. I do not know what your compressed air requirements are. As was noted, the 210 is a SMALL compressor. If you plan on running things like air die grinders, it will not push enough air. It is OK for running a 1/2" air impact wrench in short bursts, or filling car or motorcycle tires. It does not have the cfm for bigger/continuous run air tools and you can go have lunch if you plan to air up a heavy truck or tractor tire.