This is actually not true. The two languages are incredibly similar but nevertheless quite different. Quite more so than American and British English. They are different to the point where many young Czechs do not actually understand Slovak very well (well, this is a bit exaggerated, however I was surprised by kids that were just staring at me until I started speaking in Czech when asking for directions). Slovak people tend to speak Czech somewhat better because they are (or at least were) exposed to the language more. At least in my time no movies, books etc. were translated to Slovak.

Czech and Slovak have also quite a lot of differences in grammar (for example Czech have one more case in their declination system).

I tried to find how different they are, is this "virtually identical"?

https://en.wikipedia.org/wiki/Comparison_of_Slovak_and_Czech

Apparently "present-day standard Slovak wasn't codified until the 19th century." Then it's not something invented after the breakup.

They have different letters. It's the equivalent of British English lacking the letter "h" and American English not having "ng" as a digraph.