A VFD is simply a rectifier that converts the incoming AC current into DC current. Then 3 banks of 2 power transistors that fire in sequence to generate 3 phase AC current from the DC current. The VFD control board tells the power transistors when to fire allowing the controller to change the frequency of the transistors firing hence the frequency of the generated 3ph current.
Some VFD's have wiring lugs for "DC in" or "DC bus" which allows bypassing the rectifier all together and powering the VFD/motor from a large DC source.
A 3 phase rectifier is just a single phase rectifier with 2 extra diodes. Growing up our lead acid battery charger was a variac (variable transformer) with 2 legs of a 3 phase rectifier wired in. My father (an electrical engineer) had a dozen of these huge 3 phase rectifiers, I still have a few left.
1 phase rectifier vs. 3 phase rectifier
There are a few reasons I can think of that a 3 phase VFD might not work using 1ph AC in:
1) The VFD controller has voltage detect sensors on each of the 3 incoming legs and is programmed to throw an error if it doesn't detect voltage on all 3 incoming legs.
2) A 3 phase rectifier was chosen by the design engineer with diodes JUST barely big enough to supply the needed DC current and losing 1 leg (1/3 of the rectifier) puts the remaining 2 sets of 2 diodes in an over current condition that will eventually over heat them and burn them out.
3) The DC current from the rectifier running on single phase isn't filtered enough (capacitor bank) to provide the smooth DC current the power transistors need.
I have come across #1 but never #2 or #3 in my limited VFD experience.
All the really old VFD's I ever worked with were WAY over spec'd in their design and had plenty big enough diodes in the rectifier to run on single phase (only 2 of the 3 legs powered).
3ph VFD's would run a 3ph motor on 1ph WAY back when before they were ever designed or spec'd to do so. This was just never a function that was considered in the VFD design because a 1KW Allen Bradley VFD cost thousands of dollars so nobody would ever use one for phase conversion to power a 3ph lathe in their garage.
We have big cheap MOSFET and IGBT power transistors to thank for making VFD's affordable for a home/small shop use. I also believe that there are a few IC (Integrated Circuit chips) designed to provide all the control electronics to control a VFD on a single chip and most all modern VFD's use one of these few available control chips.
The power transistor network in a VFD is designed to drive an inductive load (a motor). So while it is possible that running 2 VFD's in series "might" work that really isn't what they were designed to do. Many VFD's specifically state that they are made to only power a motor and nothing else (as opposed to powering 3ph lighting, 3ph power supply, 3ph welder, 3ph VFD, etc.).
Decades ago I tried using a big huge Allen Bradley 3ph VFD to run a motor off of single phase. I discovered that the low voltage DC current to run the control logic for the VFD came from a small power transformer that was powered from legs S1 and S2 so these were the 2 legs that HAD to be connected to the 1ph AC current. Powering S1 and S3 nor S2 and S3 would work to run the VFD on 1ph.