It's been known since the US Apollo Program in the late 1960s that The Moon has magnetic rocks - far weaker than The Earth's, but strong enough for accurate measurements.
The magnetised regions are scattered across the surface rather than forming a generalised 'polar' pattern such as Earth's. In some regions, the field can be measured at around 0.3μT (micro Teslas).
Since it's widely assumed that The Moon does not have a molten iron core, the origins of the fields remain unexplained.
There are several hypotheses :
Example publication : Sci Adv. 7(32):