The 208 Volts quite likely comes from an old standard in the U.S. where transformation was done with a "Wye" connected transformer providing 208 / 117 Volts. The transformer was(is) center tapped to provide 208 phase to phase with 117 from phase to center tap, usually grounded. It is still used, though fairly rare now.
When I worked at the "Pipe Shop"(a foundry) in the '70s, it was not that common for 240 circuits, but had a strong showing with 416 / 277. Last I checked, a while back, 277 lighting was still widely available.
The motor with a 208 Volt rating is probably an older model, designed before delta-delta became the norm for distribution. With the normal swing of voltages, +- 10%, there probably won't be any problems. If you have a full 240 Volt line, it might be preferable to watch for motor overheating for a while. But for hobby use rather than production, you shouldn't notice any serious overheating.
The biggest concern with that rating is the implied age of the machine. I would watch insulation condition. That is a very old motor and the insulation may not be in that good a shape. Don't go into the motor unless you really need to. If you can get your hands on a megger, it would be a good idea to check it. But if not, don't worry about it. Just watch for trouble the first few times you fire it up.
There are many other comments on such a rating. I'm sure some will show up here over time. The bottom line though is that synthesized three phase is not a true sine wave, and older motors have some trouble with it. Especially if you get away from 60 Hz. If you do opt for a VFD, tune the output to a very low voltage, say 210-215. You should be able to go from 45 to 75 Hz with no problem beyond a reduction in power availability from the motor(s).
You speak of a DC motor as well. If the controller for the DC motor is onboard the machine, DC can be provided with a couple of bridge rectifiers. Set them up in "open delta" configuration and be prepared for a hell of an increase in the single phase input. Over and above the 1.73 multiplier.
The DC won't be upset by the synthesized / stair-step waveform, regardless of output frequency. It simply is a matter of getting DC from the AC. The only concern there is to be sure to include that into the Horsepower rating of the VFD.
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