what is the Sovereign Military Order of Malta? and why are they able to issue diplomatic passports?

Answer 1

The full official name is Sovereign Military Hospitaller Order of St. John of Jerusalem of Rhodes and of Malta (in English) or Sovrano Militare Ordine Ospedaliero di San Giovanni di Gerusalemme di Rodi e di Malta (in Italian). Conventionally, they are also known as the Order of Malta.

A history is given in Wikipedia (convenient link below). The are recognized by the UN and considered an independent, legitimate entity. Curiously, it has ...
" no sovereign territory since the loss of the island of Malta in 1798. The United Nations does not classify it as a "non-member state" but as one of the "entities and intergovernmental organizations having received a standing invitation to participate as observers"."

As such, it issues diplomatic documents which are accepted by most countries.

"SMOM has formal diplomatic relations with 96 states (many of which are non-Catholic), and has official relations with another 6 countries, non-state subjects of international law like the European Community and International Committee of the Red Cross, and a number of international organizations. Its international nature is useful in enabling it to pursue its humanitarian activities without being seen as an operative of any particular nation."

Answer 2

Malta is a soveriegn country even when its territory is small. Thus, its government can issue passports for its citizens.