The "first societies" to exist were not the first societies to use currency by a long shot. Despite some impressions from modern society, especially moder post-colonial society where specific, largely European cultural norms have been adopted worldwide, money is not a universal trait in human society. Coin currency-based money economies didn't exist anywhere until the First Millennium BCE, when it was invented at least three times independently.

I should first address cowrie shells. The most famous use of cowrie shells as a pseudo-currency today is probably from 16th Century CE West Africa. Cowries, the shells of large sea snails are relatively delicate and rare and generally considered visually appealing They're easy to carry around, so they were used as a convenient and widely accepted medium of exchange in West Africa prior to heavy European involvement. European traders, especially slavers, realized that, while rare in West Africa, cowrie shells were signficantly more common in the Indian Ocean, especially the Maldives. They began importing cowrie to exchange with West Africans because the shells were practically worthless to the Europeans despite being valued locally. The sudden influx of shells effectively tanked the West African cowrie economy.

This illustrates one of the key reasons we cannot consider cowries a true currency. As naturally harvested shells, they are impossible to standardize and control even on a local level. In that regard, trading cowries was still fundamentally a barter system, but they can be compared to currency because they have no practical value beyond being widely accepted as a store of value.

However, 16th Century West Africans were not the first people to use cowrie this way, though they may have been the people who used them most widely and consistently. There is some limited evidence of cowrie being used in a similar way in Ancient Egypt, China, and India. Like West Africa, these were all places that were either geographically removed from cowrie producing snails or with only a small cowrie snail population. In each case, the use of cowrie as a medium of exchange faded out as they became more connected with other regions and cowrie either became more common or less valued than other acceptable options.

The debate really comes down to which one should get credit for which one evolved into an actual currency, where part of the value was derrived from standardization and government support in addition to its worth as an actual resource.

1. China - archaeological work has found bronze models of cowrie shells in Yin and Anyang dating to c. 900 BCE. Knowing that cowries were a medium of exchange, we can guess that bronze shells were intended to demonstrate value or be exchanged, but lacking standardization its hard to identify these as a true currency. One piece of evidence that these were not currency in a modern sense might be that these bronze cowries were sometimes covered in gold leaf starting in the Zhou Dynasty period. That would distort the apparent metallic value, let alone any standardized expectation, suggesting they may have been more of a display of wealth than an actual medium of exchange.
   1. That would push the first use of standardized bronze coins, often in the shape of specific day-to-day objects like spades or knives, to the Warring States Period, c. 400 BCE. These coins may have evolved out of popular use of bronze cowrie shells, and other bronze tokens, as a popular medium of exchange for their simple metal value. IE the coins were a codification of an existing ad hoc system that developed outside of the bounds central authority sometime between c.700-400 BCE.
2. India - Indian currency may be the example that developed most intentionally. Estimated dates range from 1000-400 BCE, but the lite consensus seems to be around 600 BCE. Earlier Indian coins are called karshapana, which were rough shaped pieces of silver, punched or cut out of a sheet of metal to create single units that weighed approximately 1 karsha, or 2.7 grams. Each coin was then stamped by a local official with various shapes that verified their authenticity. Despite their initial purpose of just creating a conveniently small, pre-weighed piece of silver, government verification and production made it a short leap to evolving into a true coin currency.
   1. Based on their constuction - cut from sheet metal - they seem to be a natural evolution of hacksilver. "Hacksilver" is a German word originally coined to describe the medieval, specifically Norse, practice of just breaking off a piece from a larger silver object to exchange it for pure metal value in transactions where the whole object was too valuable. Though originally used to describe northern European practices, similar practices appear all over the world in all time periods.
3. The Aegean Sea - Roughly at the same time that Indian city states were creating the first karshapana, several different local rulers around the eastern Mediterranean Sea started issuing "coins." The exact purpose of these earliest "western" coins is not fully understood. Traditionally, the very earliest coins are attributed to King Alyattes of Lydia, in western Anatolia, who produce small tokens made from electrum. An alloy of gold and silver, electrum was easily found by panning in several Lydian rivers but difficult to refine into its constituent parts and thus difficult to ensure a predictable gold-silver ratio in each token. One popular theory is that they were just signs of royal favor, but were handed out in such numbers that they became an easy medium of exchange in expensive transactions anyway. Alyattes' son, Croesus, ruled when the process of separating out the gold and silver from electrum first developed. By his time, these proto-coin exchanges were common enough that he intentionally minted coin currency called croesids.
   1. At the same time that royal tokens were developing into coinage in Lydia, several Greek cities in Ionia and the neighboring islands produced similar tokens made of silver, which was more common in Greece and the northern Aegean coast. These may be an extension of an even earlier tradition. The very earliest archaeological layers at the Temple of Artemis in Ephesus contain coin-like tokens apparently used or given to the Temple in the 8th Century BCE. By the earlier 6th Century BCE, similar tokens began circulating in public, once again apparently not intentionally produced for exchange but eventually becoming common enough to use them that way. From there, Greek city states began producing coins for the purpose of exchange, with different local governments becoming more trusted and more valued by merchants than others.

So we see in all three of the earliest examples of currency that coinage evolved from pre-existing patterns of exchange (ie trading small bits of metal, often created in a reliable pattern by some authority, for their value as metal). Governments then recognized the value and prestige in standardizing and controlling these patterns and minted true coin currencies to exert greater influence in the market. That was not a new phenomenon. Many the systems of weights and measures used to gauge the value of the raw metal had all developed out of earlier states trying to exert similar controls and regulations to ensure that all trade was being conducted according to agreed upon standards as well. Those systems of weight, especially in Greece and Lydia, dated back to the Bronze Age.

The first government to issue coins in the manner described in your question (Here's some money, go use it as money) could arguably be either the Achaemenid Persian Empire, or arguably several rebel rulers within the Persian Empire. When the Persians conquered Lydia, the mint in the Lydian capital continued putting out croesids as it had before, but under Darius the Great the croesid was replaced by a pair of Persian coins called darics (gold) and sigloi (silver). They do not seem to have been intended for widespread use. From about 515-350 BCE, Persian imperial coinage was still mostly focussed on Anatolia and exchange in Greek markets. In the 4th Century BCE, local mints creating coins on behalf of local governors began popping up in the surrounding region, eventually spreading down into Phoenicia and Egypt, often to started to facilitate revolts but allowed to continue after the revolts were defeated.

Only just before Alexander the Great's conquest did a Persian mint open in Babylon, apparently responding to the same familiar pattern. With Greek and Persian coinage circulating in the west and Indian coinage in the east, coins of all sorts made their way into the Persian imperial interior and became a common method of exchange across the empire. In the areas closest to Greece, Greek coinage was so common that they were functionally a part of the Greek money economy even though the imperial government wasn't officially supporting it.