By the time playing card decks had become prevalent in Europe, the Kingdom of Sicily was the Korea of its time: There were two political entities claiming to be the real Kingdom of Sicily. This reminds us, that there was no unified Italy at the time. The old kingdom of Sicily split in 1282, and it wasn't until 1816 the two claimants reunited into one kingdom, the strangely named Kingdom of the two Sicilies. 

North of these two rivalling southern countries, the Papal States (Holy See) were located. Today, the Holy See oversees the world's smallest country, the principality of Vatican City, consisting of a few city blocks in the middle of the city of Rome, Italy, but in the Middle Ages, the land-claims of the Papal States were quite substantial, running from the west coast (Tyrrhenian Sea) to the east coast (Adriatic Sea). The bishop of Rome had become a head of state in 756, after a Frankish army defeated the Lombards, but the 14th and 15th centuries eroded some of the Pope's secular political power. From 1305 to 1378, popes resided in the Avignon enclave surrounded by Provence, and, in his absence, local aristocracy strengthened their secular power  over city states such as Ferrara, Modena and Reggio (House of Este) and Rimini (House of Malatesta). Another semi-autonomous city-republic formally under papal rule was Bologna, significant for its old university, founded in 1088 AD. The ecumenical council of Basel had begun its first session in 1431, but moved to Ferrara in 1438. Its purpose was to mend the schism between the Eastern Orthodox church and the Roman Catholic church. 

Tuscany, north of the Papal States, had fallen apart in 1115, at the death of Matilda, Margravine of Tuscany, and was replaced by several long-lasting city-republics, the most important of which were Florence, Lucca and Siena. The Republic of Pisa was absorbed by the Republic of Florence in 1405. The House of Albizzi and the House of Medici were influential in Florence. Cosimo de' Medici, who was an aristocratic banker, returned in 1434 to Florence from exile in Venice and expelled the Albizzis. The ecumenical council of Ferrara moved to Florence in 1439, in order to avoid bubonic plague, and lated until 1445. The presence of learned Greeks in Florence made Greek literary works available, such as Corpus Hermeticum, the dialogues of Plato and the works of Plotinus and Proclus. Cosimo de' Medici attended lectures by the Byzantine neopagan philosopher Gemistos Plethon (1360-1454). After decades of intermittent war with Milan, Florence entered a period of peace when its neighbours, Milan and Venice, signed a peace treaty in 1454. From 1462, Medici sponsored the Florentine philosopher and translator Marsilio Ficino (1433-1499), who gathered a circle of thinkers, the Florentine Academy, around himself. One of the members was Pico della Mirandola (1463-1494), one of the first Christian Kabbalists and a cousin of the poet Matteo Maria Boiardo (1441-1494), the latter famous for his chivalric epic Orlando Innamorato. Sandro Botticelli was born in Florence in 1445. Leonardo da Vinci was born in the Republic of Florence in 1452. Michelangelo was born in the Republic of Florence in 1475.

Milan had been a city-republic, governed by several wealthy families, among them the House of Visconti and the House of Sforza, but became a duchy in 1349. A republic was proclaimed at the death of Duke Filippo Maria Visconti in 1447, but  in 1450, Francesco Sforza (married to Bianca Maria Visconti) dropped all pretense of governing a republic, and proclaimed himself Duke of Milan. He defeated Milan's eastern neighbour Venice in the following years.

The Most Serene Republic of Venice was a very old country, founded in 697. Yes, 697 CE! Padoa and Verona became close allies after 1405. Trade with the East Roman (Byzantine) empire had favoured the economy of Venice for ages, but, when Constantinople fell to the Ottomans in 1453, Venetian economy took a hit. Intermittent naval war lasted from 1423 to 1571.

West of the Duchy of Milan, the Republic of Genoa (lasted 1005-1797) was located. It controlled Corsica 1284-1755.

North of Genoa, in a mountainous region, there were the Duchy of Savoy, the Duchy of Aosta, the Principality of Piedmont and the County of Nice; all of them ruled by the House of Savoy (much later kings of Italy).

Sardinia was under the Crown of Aragon (one of the Spanish kingdoms) for many centuries.

There were also a few dozen other small political entities, but, of those, only The County of Monaco (since 1612 The Principality of Monaco) and the Republic of San Marino (independent 301 CE) ought to be mentioned, because they are still independent sovereign countries, but, as far as I know, none of them never developed any separate card deck or tarot deck of its own.