How does the second order time constant affect circuit behavior?
Both have 0 as the steady state condition.


You are confusing transfer functions for signals.  A transfer function describes a system's behavior; it is a characteristic of some other thing.  A signal is more or less a function that evolves in time.

\$H_1\$ and \$H_2\$ both have a DC gain of five, so if the systems they represent are excited by a signal that reaches a steady-state condition, their outputs will each reach a steady-state condition five times the value the input settles to.


  If \$T_1\$ is the time constant of one of the cascaded first order circuit (\$T_1 = −\frac{1}{\tau_{p1}}\$).


Time constant implies a variable with units of seconds.  Your \$T_1\$ has units of 1/seconds.


  will this value have any meaning, ... for a first order circuit (for example, the time to decrease/reach certain value)?


Not particularly.  Google "dominant pole".  Basically, in a heavily overdamped system that doesn't have significant pole-zero cancellation, the slowest pole wins, and eventually dominates the response.