As @night cub pointed out, it was probably a privately owned cab and not part of a company by that point. That Caprice is a 1995-96 model so seeing it still used in 2003 wouldn't be that uncommon.

Hello back, surfing the internet looking for photos of these cars in the 2000's
(Yes, I am a bit obsessed with the caprice taxis) and I found this photo, you see a caprice of the 90s, dated February 2003, back you see an advertising of the movie Biker Boyz so this photo is authentic, that movie came out In 2003, what do you think?

https://www.gettyimages.es/detail/fotograf%C3%ADa-de-noticias/outdoor-view-of-morgan-stanley-fotograf%C3%ADa-de-noticias/150770017

So, if that car belongs to the taxist, he could continue using it after 8 years, if the 5-year rule came in november 1996, from 1996 to 5 years would be november 2001 and extend 3 more years to the taxi car owners until 2004, is that so? or is it what I understood, I am somewhat confused

If the cab was owned by the driver, and not leased from a cab company, there was an exception that went up to 8 years.

What shows up on Google Search: "While it approved a maximum five-year age limit for taxis, the commissioners said that they eventually intended to allow some cabbies who own their own cars to keep them as long as eight years. The retirement ages for taxis will be phased in beginning Nov. 1."

There was also extensions allowed for CNG powered cars, minivans, wheelchair accessible, hybrid or other clean-air cabs.

https://rules.cityofnewyork.us/tags/vehicle-retirement-dates

I see, then answering my question, there is probably no more 80's caprices taxis in New York at that point :(, but a friend who traveled to New York in March 2002 says that on the street there were Crown Victoria 1997 and 1996 models and 2000's models, Honda Odyssey 1995 and he found a 90s chevy caprice, he doesn't remember what year the model was but he says he was sure it was a 90s caprice
so maybe that model was still around
but in minory

PD: I have realized that for some reason my post doubled, sorry if that bothers people in any way, my intention was to ask only one time

I think that one little section of the clip was stock footage. The newest model I saw with the Caprice was the 1995 Ford Explorer and a 1993-94 Ford Crown Victoria. Also a 90s Toyota Camry was there so that part was probably filmed sometime between 1995-01

I've seen other footages videos of this guy and the other videos has the right date and place, maybe he made a mistake?
or maybe was it one of the few that remained and it was allowed to use?
since it think there was a checker taxi that stopped being used in 1999 right?

i think its footage of various era/places. At 2.13 it looks like we're in Washington DC

Also if you see the video you can see the familiar ford crown victoria that is still used today i think they circulate in nyc since 1998 right?, and also you can see a 96 or 97 ford crown victoria so i think this was in 2002 since the last mentioned model was still around in early 2000s but i wait someone who knows about nyc taxis clarify this

Hello, This is the video: https://www.pond5.com/es/stock-footage/110358919/usa-heatwave-temperatures-new-york-and-washington-shoot-reco.html , i think its not any spam just a footage video, i think i see some cars from early 2000's but judge yourselves, the video is about the 2002 heatwave in new york date april 17, in a part of the video you see these two caprices specifically minute 2:04