A double planet is usually defined in astronomy as two planets that share an orbital axis outside both planetary bodies, and by that definition the Earth-Moon system is not a double planet. However, according to Wikipedia, the “Earth-Moon system is sometimes informally referred to as a double planet (masses are roughly proportional to volumes, not surface area).“
The Moon is very large in relationship to Earth and amazingly the ratio of the mass of our moon to the mass of the Earth is larger than any other major planet and natural satellite in the solar system. This is the argument for the Earth-Moon system to be called a “double planet system.”
When I said above about the definition of a double planet system is usually defined to share an orbital axis outside either planetary body, that does not describe the Earth-Moon system. In our Earth-Moon system the axis of the orbit is well inside Earth itself.
From this perspective the issue is unresolved. Within the International Astronomical Union there is yet to be a formal definition of double planet. However, individual scientists have their own ideas and some say Earth-Moon is not a double planet and some say it is.