A payment processor normally sits between the buyer and the merchant, holds the money for a while, and can decline to pass it on. BTCPay Server removes that position from the diagram rather than filling it with someone friendlier.
The software is MIT licensed and the repository dates from August 2017. Its own foundation announcement describes it as starting as a one-man project by Nicolas Dorier. The documentation states the design plainly: it is non-custodial, payments go directly to the merchant's own wallet, a new address is generated for each invoice so there is no address reuse, and the merchant's private keys are never needed to receive. There are no subscription costs and no transaction fees, because there is nobody to charge them. It runs on-chain and over Lightning, and it is self-hosted, which is the trade being made: the merchant takes on running and backing up a server, and in exchange no third party can freeze the account.
The organisational picture is deliberately split, and it is the part most directories get
wrong. btcpayserver.org states that BTCPay Server is an open-source project and not a
company, and the contribution docs describe it as built and maintained entirely by
volunteer contributors. Separately, the BTCPay Server Foundation was announced in September
2019 alongside its first grant, from Square Crypto, to collect corporate support. The
announcement is explicit that the Foundation is not meant to control development, monetise
the software, or claim ownership of it, and its about page calls it a non-profit
organisation that will not commercialise any product and supports itself entirely on
donated capital, with Dorier holding two votes for the purpose of breaking a tie. No
jurisdiction, incorporation or legal form is published for the Foundation anywhere on its
site, so structure here is recorded as undocumented rather than guessed.
Its current listed supporters include Spiral, OpenSats, Tether, the Human Rights Foundation, LunaNode, Wallet of Satoshi, Coincards, IVPN and Unbank. Alongside Dorier, the announcement names r0ckstardev, pavlenex and Kukks as part of the distributed team.