No, private property, which is a capitalist institution, has existed since prehistoric times, probably from even since before Homo Sapiens arose. Proto-property-rights are also observed in the Animal Kingdom, with principles like "first possession", where the first animal to the kill is less likely to give up the kill, and the second animal more likely to give it up, often governing animal interactions. In fact, game theory experiments demonstrate that observance of this proto-property principle reduces conflict among agents, so it's unsurprising that it's common among animal species.
Returning to humans, extant hunter gatherer groups observe property rights, both at a personal level, with their ownership of tools and clothes:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2999363/
>>Moveable material property, such as tools, clothing, and valuables, is generally treated as individual property
And on the inter-tribal level, with tribes asserting exclusive control over the territory that they exploit.
Food sharing is common in hunterer gatherer societies, but almost all of it is intra-tribal, meaning to relatively close kin. Even within the tribe, such sharing is heavily biased by kin selection.
Food sharing is often very reciprocal as well, with an expectation of a return in exchange for the 'gift'. This is also demonstrated to some extent experimentally:
http://rspb.royalsocietypublishing.org/content/279/1740/2930