American, Schaefer and Budenberg was in Brooklyn, NY. I know they were history well before WWII. Schaefer & Budenberg was a German firm with their main offices and plant in Magedburg-Buckau, in pre WWII Germany. They had an office and plant for the manufacture of pressure gauges and related items such as your gauge calibration set located in Brooklyn, NY. The really old Schaefer and Budenberg gauges made in Brooklyn will not have the word "American" included.
I would guess your gauge calibration kit was made post WWI, given the name change to "American, Schaefer & Budenberg", and given the type case the instrument is in. Prior to WWI, I think the name would have been simple "Schaefer & Budenberg", and the case might well have been made of hardwood.
As to value, that is hard to pin down. To someone like myself who has need of a pressure gauge calibration set, it is worth something as it is a useful item. How much is another matter. To a collector of vintage engineering instruments, this gauge calibration set is worth more since it is in pristine condition. The truth is that, if this device were to be used, the pump would have to be filled with oil in order to pressurize the gauge being tested/calibrated as well as the reference/test gauge. Afterwards, putting the pump back into a velvet lined box could create a bit of a mess, no matter how well the instrument was wiped clean. As Joe in NH notes, the deadweight type of gauge tester/calibration device is preferable and about as close to "guaranteed dead-nuts accurate" as it gets. This portable device can verify that a pressure gauge is/isn't reading relatively correctly, but is only as accurate as the test gauge.
Looking on eBay, steam engine indicators seem to be listed at what seem ridiculously high prices by people who usually have absolutely no idea what an engine indicator is used for. Any of the old instruments of this type in fitted cases with nickel plating over fine workmanship seem to at least be listed for very high prices. Whether they fetch those kinds of prices is another matter. I'd throw a WAG (engineering term for "wild-ass guess") that the instrument you have might be worth 50-200 USD.
Again, this is dependent on the buyer and whether they are 'collectors' (who treat these things as museum pieces) vs a 'user' (someone such as Joe in NH, or myself, who would actually use the instrument to test/calibrate a pressure gauge).
I was born and raised in Brooklyn, NY. I came in at what I call 'the sunset of an era', when there were still many working machine shops and manufacturing industry within NYC limits, and when there was still a good bit of steam power in use within NYC limits. I have quite a few Schaefer and Budenberg pressure gauges, some from the engineering school I graduated from, when they got rid of the old thermodynamics lab. Some came from old industrial plants. I was always curious as to where Schaefer and Budenberg had their plant in Brooklyn, NY, but never could find that out. A spin-off from S & B may well have been the Amthor Instrument Company of Brooklyn, NY. Amthor continued the manufacture of dead-weight pressure gauge testers and mechanical tachometers into the 1970's in Brooklyn. Amthor also vanished into thin air and I cannot find a street address for them. I would not be surprised if S & B and then Amthor were connected, and possibly located in the Ridgewood or Bushwick areas of Brooklyn. These areas were German strongholds, and a likely place for a firm with German ties and employing fine machinists and toolmakers to be located.
The portable pressure gauge tester the OP has looks like it was never used. Had it been used, there might well be wrench marks on the nickel-plated bands on the elbows, and might also be matting of the velvet case lining from oil. The old Schaefer & Budenberg logo on the gauge face will have an oval with a pressure gauge, and a 'cell' such as might have a metal diaphragm, below the gauge. Really old S & B stuff will have "Magdeburg & Buckau" in that logo, a reference to S & B's German headquarters.