I want to summarize what I read on this topic. I did a lot of research and found some interesting things. Possibly you will find my analysis interesting. Comments with opinions are very welcome.
I first tried to find the etymological origin and the original meaning of the preposition у and opened the etymological dictionary by Preobrazhensky. Here is what he says:
У предл. c P.; звачения oт, во власти, при, близ, около; напр., просить y кого-нибудь; «у порога наш супостат». ... Значение предл. y во власти развилось из преф.; напр. «спроси y отца» звачит «от отца того, что есть во власти отца» — и т.п.
This source clearly says that possession was a derived, not original, meaning.
To verify this, I talked with a Polish exchange student to find out what the same preposition u means in his language and whether this preposition can be used to express possession in Polish. He gave me two example sentences:
(1) Mam kota.
(2) U mnie jest kot.
He explained me the difference between the two sentences. Sentence (1) means, "I have a cat," "I own a cat." The sentence says that the cat is mine. Sentence (2) means, "There is a cat at my place." It is not necessarily my cat. It well may be a stray cat who came to my place to eat something. The cat is just close by.
Asked how he would say "this fish has sharp teeth" in Polish, he said, "Ta ryba ma ostre zęby," and added that it is impossible to express the same idea by means of the preposition u. He said that an attempt to do so would sound "fake and stupid" despite that it perfectly works in Russian: "У этой рыбы острые зубы."
The explanations by the Polish student in conjunction with what Preobrazhensky says made me pretty confident that possession was not the original meaning of the preposition у. The original meaning appears to be proximity or rather "somewhat displaced from."
I also recalled what I had heard about the Indigenous Australians: They lived in tribes, where every single thing belonged to the whole tribe, and their languages did not develop general verbs of possession akin to "have." I quote from a research article:
Kayardild, a Tangkic language spoken until recently by Kaiadilt people of Bentinck Island in the Gulf of Carpentaria in northern Australia (Evans 1995, 1998), has few verbs of possession and no general verb of possession akin to “have”. The verb karrngija, “keep, keep hold of”, frequently used in discussion of country, can mean “look after”, “guard”, and “be responsible for” something in one’s possession (Evans 1995:55). Yolngu use the word dja:ga (“look after”) in a similar way, as well as ngayathama, “hold”. ... The Kayardild language has no general concept of “property” (Nicholas Evans pers. com.). The same may be said for Yolngu dialects. (Source)
So my wild imagination shows me the following scene. Imagine a Slavic tribe, where every single thing belongs to the whole group. One man, Dobrynya, asks his tribesmen, "У кого сейчас большой топор?" ("Who is the big ax close by?") He gets an answer, "Наверно, у Пересвета - он вчера им рубил дерево." ("Probably Peresvet - he was seen using it to cut a tree yesterday.") Dobrynya goes to Peresvet and takes the ax in order to use it. Now the ax is "у Добрыни." Neither Dobrynya nor Peresvet own the ax - the ax merely was "у Пересвета" and now is "у Добрыни." Dobrynya will use the ax and keep it until someone else takes it to use it. The ax is in collective ownership, and Dobrynya is now the temporary ax keeper, the current holder. He will keep the ax close by - "будет держать у себя."
So I hypothesize that the form "у меня," originally meaning proximity, was also used to express keeping a tool. After all, the tool was kept by its keeper close by.
It is a matter of fact that for some cultural or historical reasons the Russians, unlike most other Slavs, failed to deeply incorporate the concept of ownership into their language and, in particular, to develop wide usage of verbs of possession akin to "have," so the form "у меня" survived and is nowadays used to express both proximity and possession.
What could the reasons be? I studied the history of Russia in detail and see some possible factors. I will list them below.
The Russian culture is highly collectivistic. Until the very last centuries, Russian villagers tended to live in extended families, where many things were considered belonging to the whole family. Even if a certain thing was de facto used solely by a single member of the family, he was probably highly reluctant to say something like "I own it," because such a statement could be interpreted as "I will never give this thing to anyone" and make others very angry, especially the head of the extended family, who could see this as a challenge of his authority and respond, "You own nothing here." To avoid social conflicts and avoid being deemed selfish, people apparently had to use the most neutral form - "у меня."
Unlike most other Slavs, the Russians were for a few centuries under the Mongol-Tatar yoke, which substantially affected the development of the Russian culture in many ways. The Mongols and Tatars were nomads and had values and mentality very different from those of the Russians, who were settlers. The Mongol mentality was: Everything belongs to the Great Khan and everyone is his servant. During the period of the yoke, no Russian really owned anything in Russia: The Mongols and Tatars chose temporary land managers from noble Russians and gave them temporary permissions to manage various Russian lands - "ярлыки на княжение." At the very same time most other Slavs were developing their settler cultures in a natural way, not being affected by nomads, and therefore were in a much better position to develop the concept of ownership and integrate it into their languages.
For most of the time after the Mongol-Tatar yoke, Russia was characterized by a strong "power vertical," with absolute power of the tsar within the entire tsardom as well as with absolute power of land managers within their lands. This more or less resembled the social structure of the Mongol Empire. The land managers were called "помещики," from "помещать" ("to put"), and were "put" on their lands by the tsar, who could later "put" them on different lands.
Serfdom was abolished in Russia very late, in 1861. Before the reform, an overwhelming majority of Russians could not own anything, simply because of being serves.
In 1917, just a few decades after the abolishment of serfdom, Russia became a communist country, where the concept of ownership again became almost non-existent. People worked in "колхозы" (collective farms) and lived in state-owned flats, many of which were "коммунальные квартиры" shared by a number of families. No one owned lands or factories. I think the communists strongly disliked the verbs of ownership like "иметь," "обладать," "владеть" and may have discouraged their use in schools, newspapers, on radio, and so on. It is only in the 1990s that Russia ceased to be a communist country.
In view of such history it is not surprising to me that the concept of ownership has not yet been deeply integrated into the Russian language.
So my short answer is this: Apparently, the form "у меня" originally meant proximity and then was also used to express keeping a tool, and this form survived and is nowadays used to express both proximity and possession, as the Russian language did not deeply incorporate the concept of ownership and, in particular, did not develop wide usage of verbs of possession akin to "have," with some of the possible preventing factors being the Mongol-Tatar yoke, late abolishment of serfdom, and the communist regime.