Commonmode voltage and measurement with voltmeter and an instrumentation amplifier
Your voltmeter is floating and battery operated. The same cannot be said for the instrumentation amp. I have measured up to 32 KV with an instrumentation amp, but with 1 gigaohm probes that divided the input by 10,000.
I would not say that a voltmeter is inferior. It may not be as fine tuned as a instrumentation amp can be. Since it 'floats' using battery power it rejects common mode voltage and noise.
SGND and AGND are basically the same thing. One is signal ground and the other is the analog ground ref for the instrumentation amp. For accuracy and common mode rejection it should have a short path to SGND, using wide traces.



  At Figure 3, I cut the connection between SGND and AGND and I read
  this causes bias currents not to return to the source and the
  instrumentation amplifier will not work correct. But why doesn't this
  issue appear in voltmeter's case even though it also has the same way
  of 2-wire connection.



Once again by cutting the gnd link, SGND and AGND, you are letting the common mode input 'float' to whatever voltage it wants to, especially with high impedance instrumentation amps combined with AC outlet leakage currents, mostly capacitive.
Once again the meter is battery powered and floats better because it has no other ties to the circuit. There are no spurious noise or currents with a hand held meter. The reading it gives you is the truth.
What can dominate AC outlet powered equipment is capacitive leakage by conventional transformer or by a switch-mode transformer. Capacitive leakage can be 100 pF or more. It is not a shock hazard as the leakage current is normally 100 uA or less and 50 uA or less for medical equipment. However instrument inputs can detect such leakage either as a common mode error, which should cancel out if the same for both (+) and (-) inputs. If the inputs do not have a common signal ground return of the same impedance then errors can get into the readings. Open (float) the input probes to detect errors that should not be there. Short them together to check for DC offset errors. Short them to signal ground to check for common mode errors.
Sometimes with AC powered equipment an isolation transformer helps with jitter and other unbalanced common mode noise, by adding another layer of isolation from the AC outlet. Use a good DVM to compare instrument probe noise or DC offsets to Earth ground. If a signal source does not have balanced impedance for both inputs, errors can occur. Jitter is a sign of noise getting into the instrument as an unbalanced signal.



  But in Figure 2 the amplifier is in danger because of the common mode
  voltage.



Instrument amps are often protected by high value resistors and common ground resistors that divide the input by 10:1 or 1,000:1, then set the gain of the amp to make up the loss. This protects them when measuring high voltage, either single ended or as a common mode voltage where you measure current flow.
For that reason it pays to buy top grade DVM's like the Fluke 87 III series, and do yearly calibrations, to check against the instrument amps. You can question wall powered circuits common and differential stray voltages and currents, but not a precision DVM.