Slovak Rail website is correct, anyone travelling in a single or double compartment within Slovakia needs 1st class ticket. This is different to international travel, where 2nd class ticket for double is sufficient.
Btw, if you are ready to play a small trick, then it is possible to save in such situation anyway. Just purchase 3 sleeper reservations for the same compartment (2/3, Tourist), it will cost you app. same price as 2 reservations for double. Afterward at another ticket counter - two 2nd class tickets. When boarding the train just hide one out of three reservations and show conductor only two 2nd class tickets and 2 sleeper reservations.
Nice trick, but the problem is that some cashiers are not willing to sell just reservations without buying the underlying tickets (or at least showing the tickets if they were bought in advance).
... and you don't need any identity card to buy them. Or you can buy all those tickets and reservations on-line.
Hmm... maybe it does happen, but, as far as we know, there is no such railway rule in general. Does it?