Translation:Does the Netherlands have a prime minister or a president?
He is not president in the meaning of "head of state" - which is the meaning that is relevant here.
He just holds an office that contains the title "president": "Presidente del Gobierno de España" - "President of the Government of Spain", not "President of Spain". That office is called Prime Minister, First Minister, Chancellor... in other countries. Of course the title "president" is used in other contexts too, e.g. there may be a president of the first chamber of a parliament, at the same time a president of the second of a parliament, a president of the local chapter of the national stamp collectors union.... but in this context we of course are talking about the position of "head of state", and that Mr.Rajoy definitely isn't.
Of course in history there was a lot of misuse of political terms (e.g. calling the elector monarchy of Poland a "republic") but this is not a case of that.
Wow, catty much? Landsend is actually correct. Spain doesn't have a president, they have a prime minister. If you go to Google and type in "President of Spain" it redirects to results for "the Prime Minister of Spain." Just because the beginning of his title says "presidente" doesn't mean he's actually the president of the country.
It's just that the construction you suggest is not correct in grammatical terms when referring to standard English (which doesn't mean it's appropriate and normal in certain varieties/dialects, of course).
In standard English, the only correct way to phrase this are the following:
• Has the Netherlands got a prime minister or a president?
• Does the Netherlands have a prime minister or a president?