Someone else asked about various planets located at the orbital range of the Moon. It made me wonder if an object with the same density as the Moon, at a distance which would present the same angular diameter of the Moon seen from the Earth, would have the same general gravitational effects of the Moon on the Earth, given that gravity follows an inverse square law and angular dimension follow an inverse linear law. Would, for instance, a super-Earth of the Moon's density and angular diameter create the same tides as our Moon currently does? We are, for this question, discarding our Moon for the scenario; there is only the super-Earth and the Earth.
It seems to me, intuitively (i.e. without any math, let's be honest, I'm still working on that bit), that such an object would have profound effects; the center of gravity of the system would, I would think, remain in proportion to the orbital distance, but be further out in absolute unit terms (kilometers), for instance. In my head I picture an Earth orbit with wider oscillations, given the increased total mass of the system. Yet I think the experienced pull from that super-Earth would be experienced on the ground at the same magnitude as our current Moon's gravitation.
How should I be thinking about this situation? Thanks.