Why is so different the temperatures of a compressed gas compared to a liquid/gas mixture of the same fluid and they are both at the same pressure?
Generally, compression of any gas transforms mechanical energy to thermal energy and the gas warms up, until heat dissipates.

If you have ever inflated bike tyres, you would notice the air pump gets warm ( MTB ) up to hot ( high pressure road bikes ).

When air drops down by 1500 m ( from 0.85 to 1 bar ) after passing over mountain range, it warms up by adiabatic compression by 15 deg C. Warming up to 90 Deg C to get pressure 16 bar does not seems to be much.

See Adiabatic_process

A gas, with or without its liquid phase, has 3 variables: temperature, pressure and volume. You adjust 2 of them and the gas adapt to it by adjusting the 3rd one according laws of gas behaviour.

When you keep gas/liquid mixture at the given temperature and volume, it adjusts to it by reaching its saturated vapour  pressure for given temperature  by either evaporation either condensation. 

When you set a gas to have a particular volume applying a particular pressure ( by pumping it in or out ), it adjusts to it reaching a particular temperature.

For propane gas and adiabatic compression ( without heat exchange with surrounding ):

$$p_1^{1-\gamma} T_1^\gamma=p_2^{1-\gamma} T_2^\gamma $$

where $\gamma=1.13$

If we consider as initial conditions 1 bar and 20 Deg C:

$$293.15^{1.13}=16^{-0.13} \cdot  T_2^{1.13 }$$

$$ T_2^{1.13 }=   293.15^{1.13} \cdot  16^{0.13} $$

$$ t =  T_2 - 273.15 =  {( 293.15^{1.13} \cdot  16^{0.13} )}^{\frac1{1.13}} - 273.15$$

For given data it leads to about 130 deg C.

If you suddenly compress propane to 16 bar, it would get hot. If you compressed it enough and there was some oxygen, it would  self ignite. This effect  is used in diesel engines.

But if the compression of propane is gradual and gas is in contact with surrounding,  heat gets time to partially dissipate.

If the hot compressed propane is kept at constant external pressure 16 bar, it gradually contracts while cooling down.

When it's temperature reaches the temperature, at which saturated propane vapour pressure, it becomes gradually condensation.

Gradual condensation at constant pressure maintains constant temperature, as dissipative heat loses are compensated by releasing the condensation heat.