Yesterday, I saw on the news that Turkish Delight/Lokum is registered by a Cyprus Company in EU with the name of Cyprus Delight/Lokumi, hence it will be illegal to use "Turkish Delight/Lokum" phrase in EU countries.
"The period of objection is over so this cannot be undone" said Cypriot Minister of Agriculture.

This one was tried before for some other things.
I really cannot understand why they are trying to own Turkish originated foods.

Well, some details about lokum follow here;
Lokum is an Arabic word, meaning "lumps of food".
Lokma is the singular version.
A similar food was/is produced in Persia since long.
Turks got it from Persians. Then named it with an Arabic word.Thanks to Arabic influence on Turkish.
Other countries got it from Turks. Most of the countries who lived under the long gone Ottoman flag got it with the same or similar names.

Same is also true for some other foods that you can taste in Greece or Cyprus which have Turkish names in Greek pronunciation ( lokum --> lokumi ) and sometimes some people from these countries are trying to register these foods with their Turkish names under Greek or Cyprus flag.

I can't understand what their point is for nationalization of a food.

Anyway, I don't mean to start a food fight here but I couldn't stop myself saying a word against it.

Images of Lokum:
http://en.wikipedia.org/wiki/Turkish_Delight

You're false. Please check the Wikipedia article to learn more;
http://en.wikipedia.org/wiki/Turkish_Delight

There's nothing there about Persians, but you should know that Iran was also a part of the Ottoman Empire;
http://www.dartmouth.edu/~gov46/ottoman-empire-1580.gif
And the word "lokum", which is the original name for "Turkish Delight" does not come from the Persian language.

I really can't understand what Cypriots try to do, but everybody knows that lokum has been produced since the 15th century in Turkey (or we can also say in the Ottoman Empire).