If he introduced himself as Officer Dan Page, or anything that identified him as a member of the city’s police force, then he doesn’t.
Not sure about the cops but in the military there is a specific UCMJ article that is used in this regard “detriment of the service” or some such (been a while).
Your not allowed to do any political activity supporting a certain candidate or position while in uniform or identify yourself as military unless you are acting in a role that requires interaction such as a PAO or CO and then it must be apolitical.
Technically retirees are also under the same rule as they are on the retiree rolls - but they typically let it pass.
I know for the military, federal, and state employees, there are limits on what you can do with respect to speaking engagements, or other gatherings (can't wear uniform to political rallies, etc...). I'm certain there are limits at the county/city level, as well.
Court cases...labor law and others....have set out that police and fire can be disciplined up to and a including termination for speech that the employer finds embarrassing to the organization even when that speech has been conducted off duty. This includes even when the employee does not wear their uniform or identify their employment status if it is easily found out by the public that they are employed as police and fire....aka google searches.