Could an asteroid or other moving celestial body become another moon of the Earth?

It *is* possible, but it's very unlikely....

The two moons of Mars are likely captured asteroids, as are a few of the moons of Jupiter. Triton, the largest moon of Neptune, orbits backwards compared to it's other moons - and, for this reason, it's thought Triton might be a captured Kuiper Belt Object similar to Pluto.

So - we *do* see examples of captured moons; but, while it's possible - what's involved is that the body has to enter the planet's gravity field at just the right angle, just the right speed, to "fall" into orbit. (We go through a similar process when we send orbiters to Mars, Mercury, and other planets - we have to adjust their speed and trajectory so they won't go skipping by, or crash into the planet...)

So... it *has* happened around other planets, but... it's rare.

Every so often a small asteroid, 1 or 2 metres across, enters a temporary orbit around the earth and moon. It is very unlikely for an asteroid to enter into a permanent orbit. As an asteroid falls towards the earth it speeds up, and this gives it enough speed to escape. To be captured it must loose speed. It is possible for a small body to loose speed to the moon, but the orbits that are formed are wide, looping and often chaotic. these bodies only last a few years at most. before they escape, or occasionally hit the earth (or even more rarely, the moon)

Large bodies can't be captured like this.

The window for orbital insertion is really very narrow. The object would have to arrive at exactly the right angle and speed, otherwise it would either fall to Earth or swing around and back out into space. The odds against this happening by chance are so slim that it could only realistically happen if it was done deliberately. The effect of another massive object in orbit around the Earth would depend on the object's mass, and its orbit.

There are no such Earth sized objects wandering around our solar system. We've looked. A lot.

If one flew in from outside the solar system, it would surely be going too fast, and by the adds, at not a good angle for such a gravity capture.

So no, it's not possible.

Yes - but it most likely wouldn't be stable over the long term due to the moon