Bitcoin Core is the reference implementation, and it has no owner. Its own site describes it as an open source project that maintains and releases the Bitcoin client software, and as a direct descendant of the client Satoshi Nakamoto published. The first specification and proof of concept went out in 2009 on a cryptography mailing list, per bitcoin.org. The name came later: the 0.9.0 release renamed the reference client from Bitcoin to Bitcoin Core "to reduce confusion between Bitcoin-the-network and Bitcoin-the-software", and that release was tagged in March 2014.
What the project ships is a full node. It connects to the peer-to-peer network, downloads blocks, and validates every one of them against the consensus rules itself rather than trusting a server. A wallet and a graphical interface are optional builds on top. The project also maintains related cryptography, most visibly libsecp256k1. The code is MIT licensed.
The governance is deliberately thin. CONTRIBUTING.md states there is "no particular
concept of 'Bitcoin Core developers' in the sense of privileged people". A small number
of maintainers hold commit access and merge patches, and bitcoincore.org says that role
exists "by agreement of project contributors". There is no board and no membership. The
project publishes no legal entity, and none could be established from a primary source
here, so structure is recorded as Undocumented rather than guessed at.
The money sits outside the project, and this is the part most often misread. Bitcoin Core employs nobody. Individual contributors are paid by separate organisations with their own boards: Brink funds engineers through a grants programme, Chaincode Labs employs engineers who work on the codebase, and OpenSats runs long-term support grants for Core contributors. None of that funding confers authority over what gets merged. A grant buys a person's time, not a commit bit, and the review process is the same for a funded contributor and a stranger.