The short answer (in my experience) is it depends.  For some sensors,
the ECU detects readings that are out of spec and pinpoints exact
sensors, e.g. the O2 sensor in your case.  There are several things
that can fail with an O2 sensor, the sensor itself could be giving bad
readings (out of range), the O2 sensor heating circuit can fail, etc.
The ECU is able to narrow down failures quite accurately in this case
and in the case of various other electronic sensors.

OTOH, for EGR systems for example, the ECU may report a malfunction
that represents a hi/low EGR flow.  The root cause of this code could
be one of several things, e.g. a bad EGR temp/flow sensor, a vacuum
leak to the EGR components, clogged EGR passges, a stuck  EGR valve,
etc.  So in this case further investigation is needed to accurately
determine the root cause.

To your specific case:

1. Are you sure you replaced the right sensor? (not undermining your
ability, but as you said, there are 4 sensors).
2. You could have perhaps contaminated the sensor with anti-seize.
3. Check the code - is it the same P0037 that has reposted.

Cheers,
Nirav
96 Max GLE, 120k

The codes are accurate. You could have a bad connector or bad wire going to
the sensors.