Matrix is controlled by the Matrix.org Foundation (https://matrix.org/foundation). The original team who created Matrix subsequently set up Element (New Vector) in order to fund Matrix work, but we very deliberately split Matrix into an independent foundation.
Unsure what you mean by “not a standard” - all APIs of the Matrix spec have been successfully implemented at this point by indpependent developers: this makes it an open standard. Eventually, once the standard has reached maturity, we may try to contribute to W3C or IETF - or we might turn it into a dedicated standards body like W3C did for the Web.
That is exactly what I mean by not a standard, standards are usually a bit more stable in the flow of new features than I see with a specification. That all api have been independtly implemented is news to me as I have no knowledge of any server implementations that are still not heavy wip so I will retract that comment and similarily the one of the relationship with New Vector.