I know in mythologies such as Greek mythology, gods and goddesses would act as patrons. For example, Athena was the patron goddess of Athens. Did the gods and goddesses in Norse mythology have any sort of similar patronage system for certain cities or regions?
Not really, no.
xxvi (26) That folk has a very famous temple called Uppsala, situated not far from the city of Sigtuna and Björkö. In this temple, entirely decked out in gold, the people worship the status of their gods in such wise that the mightiest of them, Thor, occupies a throne in the middle of the chamber; Wotan and Frikko have places on either side....
xxvii (27). For all their gods there are appointed priests to offer sacrifices for the people. If plague and famine threaten, a libation is poured to the idol Thor; if war, to Wotan; if marriages are to be celebrated, to Frikko.
However, Wikipedia says the existence of this temple is not corroborated by archaelogical evidence. However, there is other evidence supporting the lack of a patron; we have this passage from Germania, by Gaius Cornelius Tacitus:
The Germans, however, do not consider it consistent with the grandeur of celestial beings to confine the gods within walls, or to liken them to the form of any human countenance. They consecrate woods and groves, and they apply the names of deities to the abstraction which they see only in spiritual worship.
Source: Germania, Tacitus, 9
The same source also says that they did not even really have cities (with which to have patron gods):
It is well known that the nations of Germany have no cities, and that they do not even tolerate closely contiguous dwellings. They live scattered and apart, just as a spring, a meadow, or a wood has attracted them. Their villages they do not arrange in our fashion, with the buildings connected and joined together, but every person surrounds his dwelling with an open space, either as a precaution against the disasters of fire, or because they do not know how to build.
Source: Germania, Tacitus, 16
Further, it seems that the worship sites are distributed all over Scandinavia for the same gods:
vé (ON, 'shrine'). The term derives from Germanic *wtha and designates heathen shrines and sacred places in the widest sense of the word. Even the homes of the gods in mythological poetry of the Edda (Vafthruthnismol, 51) are called vé, and the word occurs repeatedly in skaldic poetry....
Place-names reveal the wide distribution of shrines with the name vé in Scandinavia: in Sweden there are more than 80 place-names based on -vé, 18 of them found with the name of the god Ullr, 16 with Skaði, 8 with Freyr, but only 4 with Odin and 2 with Thor. In addition to this there are a large number of place-names based on -vé which use geographical elements (e.g. Visby). In Denmark, on the other hand, all 5 place names based on -vé are combined with Odin (e.g. Odense). Most of the few Norwegian place names called Ve are not compound, and in Iceland there are none whatsoever.
Given the existence of temples to multiple gods at once, the lack of cities throughout the region with which to be patron cities, and the existence of many shrines to the same gods distributed all over scandinavia, it doesn't seem as if there were any patron cities or regions to one particular god; though there were smaller shrines and geographic formations that were dedicated to one god.