Welcome to EAF
Dragonfly.
As for your question, it's actually hard to say that they have the same gravity when ignoring how because to answer your question, we have to stay within the limits of science. The moon can't have the same gravity without having the same mass and size, or without having a different mass all together. If you were to say that both were equal in gravity the moon would have to be much more dense then earth (since it's smaller) The increase in density results in an increase in mass, hence increases the gravity.
But, strange things happen when the moon becomes more massive then earth. Just the way the moon orbits earth, earth would actually orbit the moon in a similar way. They would switch their orbital patterns. Eventually, after a long long time, earth would become Tidally Locked. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner.
As for the local effects on earth, it would be easy to say that if this was the case since the formation of Earth and Moon, then life may have never started in the first place. Earth needed the tidal bulging (pulling of earth's crust from the moons gravity (which causes the tides)) in order to produce oscillating currents known as tidal streams. These tides and tidal streams caused dramatic fluctuations in salinity around coastlines which could have driven the evolution of early DNA-like biomolecules. There would also be no seasons since the poles of earth. It's possible that if things didn't happen just as they did, things would be totally, totally different. Weather patterns would change, climates would be altered, exc...
But, as you know, this is all speculation. As we find more examples in space, we'll learn more about the implications here.
