Does the Moon's magnetic field affect Earth's magnetic field?

So would the Moon's magnetic field affect the Earth's magnetic field, just as its gravitational pull affects Earth's gravitational pull for oceans?

Yes, but only slightly. Firstly, magnetic fields can superimpose, so the field at any point is the sum of the field due to the Earth and the field due to the moon.

However, the moon is rather far away (and has a weak magnetic pole strength), so the magnetic field due to the moon on Earth's surface is nearly negligible (magnetic field also decreases as an inverse-square law)

In addition, the magnetic field of the Moon may bolster or erode the Earth's field as magnets moving relative to each other tend to either lose magnetization or become stronger. But this process has a negligible effect when we take the Moon and Earth.