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.
I've been meaning to write a detailed answer to this question for a while, so here it is. The short answer would be:
We don't (and may never) know for sure, since direct evidence on how pre-Christian Scandinavian religions were practiced is extremely scant.
However, let me review some of the current knowledge we have on this topic that might shed some light on whether Norse people had a concept of patron or tutelary deities.
In his article How uniform was the Old Norse religion?, Stefan Brink gives a very nice overview on the distribution of theophoric place-names in continental Scandinavia. He provides maps for the most represented Norse gods (Frey, Thor, Odin, Ull, and Tyr), from which he draws the conclusion that
different cults seem to have been distinctly regional in many cases. [...] A regional cult of Ullinn is demonstrable for the central and western areas of Norway, and only there. Týr was obviously an important god in Denmark — probably the most important — according to the place names there. [...] This is a strong indication that the pagan ‘religion’ in early Scandinavia was never homogeneous. There were obviously regional cults of certain gods and goddesses. Ullr and Freyr had a distinct regional cult in the Svea-dominated area in Sweden and around Viken in Norway. The cult of Ullinn was a local cult in central Norway.
Evidence against a unified Norse Pantheon
Terry Gunnell, basing his considerations on archaeological evidence, place-names, and a thorough study of the available medieval literature, published a series of papers in which he argues against the notion of a Norse panthenon, in the way it is presented in Snorri's Edda, in some poems of the Poetic Edda, as well in Adam of Bremen's account on the temple at Uppsala. His conclusions are summarized in Pantheon? What Pantheon? (see references therein for more material on this subject). In particular, he argues against the idea of a coherent Pantheon universally recognized throughout Scandinavia, with each god associated to a specific function or social class (e.g. Thor=Warrior class, Frey=Fertility, Odin=Magic):
[...] there is reason to consider that many of the gods known in the Germanic and Scandinavian world were tribally-, area-, or clan- (family-) related rather than bound to social classes [...]. There seems to be little evidence to suggest that rural communities ever worshipped a range of gods (even if they had heard of the gods worshipped by other people).
In particular, on Adam of Bremen's account he says:
[...] why Þórr, Freyr and Óðinn should appear together in Adam of Bremen’s earlier-noted description of Gamla Uppsala? [...] it should be remembered that the festival described by Adam was a large-scale “national” gathering of people from different communities within a wide area, who may well have had different religious backgrounds. It was not an annual occurrence (taking place every nine years).
Tacitus's Germania and Germanic tribal gods
First of all, I would like to point out that using Tacitus's account to infer information on Norse religious practices should be done with extreme care, as it relates the customs of Germanic peoples centuries before the earliest developments of what would then become Norse culture. Nevertheless, it is interesting to note that he makes explicit mention of tribal ancestor cults among various Germanic peoples:
In ancient lays, their only type of historical tradition, they celebrate Tuisto, a god brought forth from the earth. They attribute to him a son, Mannus, the source and founder of their people, and to Mannus three sons, from whose names those nearest the Ocean are called Ingvaeones, those in the middle Herminones, and the rest Istvaeones. Some people, inasmuch as antiquity gives free rein to speculation, maintain that there were more sons born from the god and hence more tribal designations—Marsi, Gambrivii, Suebi, and Vandilii—and that those names are genuine and ancient.
The particular case of the Ingvaeones has attracted the interest of many scholars because of the similarity of their name with Yngvi, another name of Norse god Frey. Here's what Gunnell has to say on the matter in the above-mentioned article:
Place-name evidence (Brink 2007: 109–11), Saxo Grammaticus (1979: 174), and even Snorri Sturluson (1941–51: I, 24–25) all suggest that Freyr’s cult was geographically connected to the Uppland area, and the Ynglingar kings (Vikstrand 2001: 55–71; and Sundqvist 2002). There are also obvious connections to the Ingvaeones tribe (Tacitus 1948: 102), something which once again encourages consideration of whether the Old Norse gods might not have originally been earlier more tribal- and/or geographical- rather than class-orientated, and whether they, like the matronæ and dísir (see Gunnell 2005a) might not have been seen as independent figures at the head of cults which were also connected to ancestor spirits, as Philip Shaw suggests (Shaw 2011: 47, 71–72, and 99–100).
Similarly, I would like to add that also the Saxons were known to venerate a national patron god by the name Saxnot, along with the universally known Germanic gods Wodan and Thor.
To conlude, even though the scarcity of direct evidence prevents us from making strong statements on Norse religious practices, modern studies point to the possibility of Norse religion being much more regional, tribe- (or clan-) based and henotheistic than how it is presented in works like Snorri's Edda and in part of the Eddic poetic corpus. For this reason it is reasonable to expect that specific deities would play the role of protectors or patrons of specific regions, tribes or, at a smaller scale, even single clans and families.