Markets were a key part of the economic life of the Middle Ages, much like digital and global markets are in our day and age.

Alongside its obvious function of goods exchange within the immediate premises of the city or town which hosted them and the rural areas in some manner connected to the city proper, markets have always had an intrisical political weight attached to their economic and then financial outputs. The networks of relationships and dealings which came into being during the expansion of the cities and their population during the XI century in Italy for example provided the solid foundations atop which government experiences could be constructed in several fashions.

Taking as an example the city of Florence, in Tuscany, we have one of the most telling examples of industry and commerce "tycoons" (since it's an evoking term I've used it but please, take it with a grain of salt) carving out political power for themselves. After the creation of the legislative and government bodies of the city during the 1100s and 1200s, in 1293-1295 a series of laws was passed, named "Ordinances of Justice" which prohibited anyone who had a knight or noble in their family tree to access public offices, in an effort to curb the aristoracy's power against the interests of those craftsmen, merchants and bankers who were sorted to hold government ranks. Said craftsmen, merchants and bankers had organized themselves into trade organizations, each providing the demographic numbers to sort out the members of the city's administration. This procedure would be in place in the city well into the XV century and would also result in conflict within these urban middle and upper class, as in the 1383 Ciompi revolt, starring the salaried workers of the wool industry against the immensely wealthy and influencial trade magnates which at times monopolized the six offices of priori (mayors) of the city's council.

This was the case in many other cities in Italy. In the kingdom of Naples, in the south, both in Agenvin and Aragonese times (1266-1497), mayors and other public officials within the cities were elected among the citizienry. Said citiziens were often law experts (judges, lawyers, notaries and the like), craftsmen and merchants. It's telling that most of times the members of the Council of Forty and the Six Electeds of the city of Capua (the second most important city of the kingdom after its capital, Naples), were in large part merchants and bankers. Merchants and bankers are often those who contract the yearly concession of toll taxes in a town or region, or some civic offices or provide favourable loans to the city's finances in return for said contracts or concessions. These merchants were also quite important in the local administration, especially regarding economic matters, since their personal skills and training in matters of money, commercial rights and financial organization were key for the expansion and development of both trade practices and administrative practices in the ever-expanding state apparatus in place during the times of Alfonso V of Aragon and his son Ferdinand (1443-1494).

As a closing note, markets were an instrument of rulership and a political token. Markets and fairs were often regional and often said regions had developed some sort of specialization in the context of goods production (say, for example, the export of chestnuts or a specific type of fish), linked with seasonal products. Fairs and markets appear to have been born and organized to follow some specific itineraries connected to these specializations. It could happen, like in the case of the cities of the kingdom of Naples between northern Campania, Molise and Abruzzo regions, markets would occur in a given time frame, separated by some days or weeks, in order to allow foreign and local merchants to literally follow the production and exchange cycle to allow the creation of small scale economies which would attract speficic products and specific trade operators. Alongside this, the royal exemption from tolls and duty taxes during a fair or markets, total or partial, was an important political tool to reward loyal cities (often case, urban centres within the royal demesne) and to boost the economy by reducing transaction costs. It was frequent that these exemptions were distributed by taking a very close look to the aforementioned scale economies and "market chains", which also provided the legal framework and accountability networks to ensure fairness in busisness and prevent misconduct.

About the notion of towns being ackowledged as such only if possessing a market, I have encountered this claim only in historic fiction, so I really cannot say. For the other cases, I've met court cases where the court decides for a prohibition to hold market in a said location, but never I have encountered occasions where markets had to be explicitly allowed.

Sources:
Sakellariou, E. 2011, Southern Italy in the Late Middle Ages
Demographic, Institutional and Economic Change in the Kingdom of Naples, c.1440-c.1530, Brill Academic Publishing;
Senatore, F. 2018, Una città, il Regno: istituzioni e società a Capua nel XV secolo, Istituto Storico per il Medioevo;
Provero, M., Vallerani, L. 2016, Storia Medievale, Mondadori Education, Roma.