Basis of UAE's claims on Abu Musa & the 2 Tonbs

Because they have been under one for of an emarati administration, the eimarate of Shariqah and Ras Al-Khaimah, for the last 300 years. Before that they were abandoned and beofre that the Portuegese and before them the Muzzaffarids and before that local priceling and before that the Qaramitah and before that the Abbasid etc.

well UAE always claiming these islands peacefully and always asks Iran to sit and talk about it and as always Iran ignores such efforts and it even refused to go to international court to solve the matter, so clearly Iran is not doing much which probably means they have nothing to say to defend themselves.

and about the basis , there are documents proving the rights of UAE of these Islands from Iranian government and British government, one of these documents from Iran (1960s) asking Arab Sheikhs (whose Capital is in UAE now) to use these islands for a period of time.

and as most of Middle East problems, the BRITISH made two separate deals with both Arabs and Iranians and later acted like it has nothing to do with it and left everybody fighting.

so as Iranians has British Documents proving their ownership, Arabs too have Documents from the British in Addition to the Iranian once.

The Muzzafarids ruled Shiraz back in the 14th century so yes, several Iranian kingdoms did have soveriegnty over the Island at some time as did the Greeks. Iran took the Islands in the 70s because of strategic reasons and some historical claims. They tried to take Bahrain as well, quoting that it was a sunni ruled majority shia country as well as that it was once ruled by Iran, in the 1700s and before that during the Sassanids. The reason for Bahrain this time was oil. In 1971, Britain gave those countries independence and since they did not have any kind of military, they were pray for Iranian invasion, Saudi Arabia and the US did a behind the scene trade-off, Bahrain was not to be taken but the islands were and the shah was to get some help in building its army. The shah accepted and the Emaratis, who had no say in the negotiation, protested.