Unable to remove faucet handles on widespread bathroom faucet
I highly suspect that this faucet is very similar to the Kingston Brass style KB961/2/3/4/5/7/8/9 as shown in this specification sheet.
https://s1.img-b.com/build.com/mediabase/specifications/kingston_brass/574937/kingston_brass_kb969_m.pdf
The faucets are secured via wide flange nuts from the underside of the sink/vanity. Each faucet assembly has an inlet water connection and then a separate small flexible hose connection from the faucet to the mixer connection on the bottom of the spout assembly. The following picture is from the above specification sheet.

Removal of this type of faucet necessitates the disconnection inlet supply line, the flexible connection to the spout mixer, and then the removal of the flange nut on the bottom. Once the nut is removed the whole faucet assembly lifts out of its mounting hole from the top.
Note that many of the more modern quarter turn type faucets from multiple manufacturers mount as shown here.