What is the significance of enthalpy definition not under constant pressure?
There are many processes that take place at constant pressure, such as the change of phase from a liquid to a vapor for a pure substance at constant temperature.  Also, the heats of chemical reactions and heats of formation of compounds are typically cataloged at constant temperature and pressure.  So the enthalpy comes into play in both these instances.

The first law of thermodynamics for a closed system is expressed conveniently in terms of the change in internal energy of the system.  However, for an open system (in which mass is flowing in and out) operating at steady state, the first law is more conveniently expressed in terms of the enthalpy, not the internal energy.  And, of course, all industrial applications of thermodynamics involving continuous processes are operating at steady state.

Chet