Is the standby altimeter required for day VFR flight in a C172 Nav III?
You don't need the standby altimeter for VFR (day or night) but you do need it for IFR.

The Comprehensive Equipment List (section 6 in the POH) lists all the equipment required by the type certificate:


  Required items or equipment for FAA certification (14 CFR Part 23
  or Part 91)


But, even if a piece of equipment is required by the type certificate that doesn't mean it's required for a specific flight. The Kinds of Operations Equipment List (section 2 in the POH) lists the equipment that's required for airworthiness in day and night VFR and IFR:


  identifies the equipment required to be operational for airplane airworthiness
  in the listed kind of operations.


14 CFR 91.7 says that the aircraft must be "in an airworthy condition" to fly, and that's what the KOEL defines.

The reason that the standby altimeter is marked as required in the Comprehensive Equipment List is that the C172S was type certified under the old part 23 rules. 23.1311 required IFR-certified aircraft that use electronic displays - like the G1000 in the NAV III - to have a standby altimeter, either mechanical or electrically isolated from the main one (emphasis mine):


  For certification for Instrument Flight Rules (IFR) operations, have
  an independent magnetic direction indicator and either an independent
  secondary mechanical altimeter, airspeed indicator, and attitude
  instrument or an electronic display parameters for the altitude,
  airspeed, and attitude that are independent from the airplane's
  primary electrical power system


In summary:


The standby altimeter is marked as required in the Comprehensive Equipment List because it's an IFR certification requirement
Operationally, the standby altimeter is only required for IFR flight, per the KOEL