Of course they do have goverments. London has a government, as well as Vienna. The definition of a state according to the Montevideo Convention, in my opinion, is just targeted at former colonies in the Americas. For this reason it has also only been signed by States from the Americas.
They clearly have local governments that do not intend to have the final say on important financial and economic matters. Dubai's government does. The definition of a state according to the Montevideo Convention might be targeted at former colonies, but it has been used extensively and is referred to by far and away most scholars. It might not be perfect, but it's probably the best we have.
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How about the Saudi government? Do you know that in each province has its own emir/prince/ruler (whatever you call it)? What do you say about that?
There is usually a gliding scale in these things. Look at the United
States of America or the European Union. You have federal states, confederal states, unions.. It's not easy to pinpoint which is which. Belgium too is a federal state with two major parts.
Dubai is right on the edge. It may not be a state according to some, but only very just. One could just as easily consider it a state imo. And like Fred says at the beginning of this thread, many make this 'mistake'. If a lot of people are making this 'mistake' that tells you something.