In case you didn’t know, the Met Police in London looked to get a cut of all uses of police boxes in Doctor Who merch.

The U.K. Patent Office found against them, on the basis that, by now, a police box was just the TARDIS, and the Met were no Time Lords.

This is reflected in Doctor Who’s in universe accounts, where the blue box iconography is recognised as being the Doctor’s conveyance.

As in the show, so it is in real life. By official ruling, the TARDIS doesn’t look like a police box.

Police boxes look like the TARDIS.

@Tupp_ed
Doctor Who first aired in 1963, when police boxes were more common.

https://en.wikipedia.org/wiki/Police_box says:

In the show, although a TARDIS is normally capable of disguising itself to blend in with its surroundings, the ship's chameleon circuit broke down in England in 1963, and left the TARDIS seen most often in the show stuck as a police box.

as police boxes were phased out in the 1970s, over time the image of the blue police box became associated as much with Doctor Who as with the police.

Police box - Wikipedia

@Tupp_ed

There are references to the trademarks and objections and the Patent Office ruling on the same Wikipedia page, with citations.

Including one called "Time and Registered Documentation in Space".

https://web.archive.org/web/20041021192718/http://dwas.drwho.org/news/20021023-tardis

"The Met even had to pay the BBC's costs of £850"

Time and Registered Documentation in Space - Doctor Who Appreciation Society (DWAS)