Yes. And local authorities in the UK routinely have grants of arms and banners/flags of them (albeit most of them are newer and less classy than the City's).

Ditto police forces: although most smaller authorities' police have merged into larger county or larger forces, there's nothing odd in principle about a municipal police force. The City's is strange, in that it's entirely enclaved within the Met, but there are plenty of oddities out there (the Mersey Tunnels Police, say).

There are several ancient cities with 'Lord' mayors, including York.