An old-fashioned meter (e.g., Simpson 260 VOM) measures the voltage between the two input terminals by connecting a largish resistor in series with a microammeter.
"20,000 ohms/volt" means a 50 uA meter with 200 k\$\Omega\$ total resistance to make a 10 V full-scale.
New-fangled meters (the usual 3.5-digit DMM) connect the two input terminals to a switchable voltage divider to obtain, say, 200 mV at the full-scale setting of the voltmeter.
The internal ADC is scaled to obtain 1999 counts at 199.9 mV at the divider output: it's an electronic circuit.
Nothing in the usual hand-held meter (of either type) is connected to ground, and "voltage" is always the difference between two points.
At higher frequencies, consider a coaxial transmission line. Cut the cylindrical line with an imaginary plane perpendicular to the wire. At a given position along the line, the voltage is the difference in potential between the center conductor and the outer shield. The current is the flow along the wire through that plane. At high frequencies, this is a local measurement.