For the time being, 'Castle' has to be considered a part of an alternate TV dimension; I'd like to think it's part of the TV Land in which we'd find 'The West Wing'. This is because they've shown the Mayor of New York City to be a middle-aged black man. (I think his name is Bob.) In Toobworld, the current Mayor is Mike Bloomberg, as seen in '30 Rock', 'Law & Order', and the Muppets Christmas special.
The situation could always change in the future; that's the great thing about the TV Universe - it's always in a state of flux. If the mayor never makes another appearance on the show before it's cancelled, then we can make the claim that 'Castle' was affected by the timeline reboot of 'Primeval'. With that adjustment, even though we'd only have the original version to view, we could claim that it played out in the new timeline with Richard Castle and Detective Kate Beckett meeting Bloomberg at the celebrity fund-raiser.
In the meantime, it's not just the mayor that sets 'Castle' apart from the rest of Earth Prime-Time. It's also their version of the New York Ledger newspaper. On 'Castle' it's a broadsheet like the New York Times. But in the 'Law & Order' franchise, where it began, it's a tabloid like the Post or the Daily News.
The online site for the Ledger is different between the two shows as well. Here's how it looked in an episode of 'Castle':
And here's how it appeared when Jack McCoy brought up the site in the most recent episode of 'Law & Order' ("Blackmail"):
Now, if they were sharing the same TV Universe, I'd make the effort to create a splainin for the different web designs. But as it is, this is just one of the differences between the two dimensions.
The most recent 'Castle' episode, "The Third Man", had an establishing shot of the New York Ledger building:
I don't know if we ever saw it in the TV show 'Deadline', which was an off-shoot of the 'Law & Order' franchise, but again, we can use the excuse of alternate TV dimensions to splain away the differences.
Just another example of the trivial effluvia that catches my eye while watching TV....