It depends on both the Flag and Port state whether watch hour logs are mandatory. In most country’s / Flags you are not required to have a separate log of watch hours BUT, if no log is present, the inspectors are allowed to investigate the number of hours worked by watchstanders.
For example, Marshall Islands advisory 79-10 states:
During PSC inspections AMSA inspectors pay special attention to verify compliance with the STCW Hours of Rest requirement. This includes checking records of Hours of Rest being maintained on board and examination of other evidence, such as log books (including cargo and other operational records), watch schedule rosters posted in conspicuous spaces and interview of watchkeepers during the audit.
US requirements are stated in 46 CFR § 15.1111 - Work hours and rest periods.
So I don’t know which flag & port states you fall under but for a Marshall Islands vessel in US waters a watch log is not required BUT if you do not have one inspectors may check “other evidence”. So basically your captain has the right to make you fill them out and, while only he knows his own intentions, it is probably to make inspectors happy and keep them from digging into “other evidence”.