Note that a wooden mortar is first and foremost a wooden utensil and the same care instructions as for other wooden equipment apply.
So oiling it regularly with a food grade oil is a good idea. You can find Q/As on how to oil e.g. cutting boards here on the site.
For general cleaning, it depends on what you used it for.
Dry herbs and spices are easy: a quick wipe with a dry cloth should suffice in most cases. Wet preparations (think curry paste) are trickier, but a rinse with hot water is your first step. You will often find warnings about using dishwashing liquid and they have a point. Wood is prone to absorb whatever liquid it comes into contact with, and who wants the next preparation to taste like dishwater?
But your curry will also leave a taste. A common remedy for lingering smells is rice: grind it in your mortar and it should absorb it. Repeat until it remains neutral. Similar advice uses salt. And of course being diligent about oiling should reduce the absorption of liquids and flavors.
To be very honest, I would probably do two things:
- Reserve it for dry ingredients or at least roughly the same flavor profile - prevents that touch of garlic in your dessert - and live with the fact that it will develop a patina of past preparations.
- I might rarely even ignore the advice and use the tiniest bit of a non-scented dishwashing liquid for a very quick wipe followed by a good rinse off and probably oiling after it’s dry - but only if it’s absolutely necessary. Proceed at your own risk.
The absolute worst thing you could do is running it through your dishwasher or giving it a good long soak. Do “the opposite” and you should be fine.