What is the 'sense' pin on an op-amp?
The sense pin in a voltage regulator is connected to the point-of-load so that a high output current and wiring resistance do not compromise the voltage regulation.   The INA137 sense pin serves a similar purpose, allowing
the amplifier output to be sensed, for feedback, at an output node which might not be
directly wired to the amplifier 'output' pin. see figure 5

The matching of the internal resistors is better than easily achievable
with discrete components, so a variety of precision circuits is possible
with this kind of packaged op-amp/resistor array.   Someone chose 'sense'
as a descriptive label for the low-R inverting input connection, possibly
because it can replicate the function of voltage sense wiring.