Is a wooden watch durable enough for daily wear?

For light everyday use, yes. Wood handles knocks less well than steel and can crack or warp if it gets soaked or left in direct sun, so keep it dry and store it away from heat.

Can I wear it in the shower or while swimming?

Don't. Wooden watches are generally not built for water, and moisture swells and stains the wood even when the movement survives.

What movement does it use?

It is a quartz watch, so it runs on a battery rather than a winding mechanism. The listing does not name the movement or battery type.

How do I care for an ebony wood watch?

Wipe it with a dry soft cloth, avoid water and chemicals, and re-oil the wood occasionally with a tiny amount of mineral oil if it looks dry.

It works as a gift for someone who likes natural materials and unusual accessories. Check the case size and strap fit first, since wood links are harder to adjust than metal ones.