Learn Croatian: Putnik ide na stanicu. - The passenger is going to the station.

Because the passenger is just making his way to the station, likely on foot. The verb ići (he/she goes = ide) is the general word for going somewhere. You use voziti se (to ride/drive) or putovati (to travel) when the person is actually making a journey inside a vehicle.

If ići usually means walking, can we say he is taking a taxi to the station?

If you want to specifically say he is riding in a vehicle to get there, you would say vozi se na stanicu. However, ide na stanicu is often used as a general statement of his destination, even if he happens to be taking a cab and you just aren't focusing on the vehicle.

Is the word putnik related to the verb putovati (to travel)?

Yes! Both come from the noun put, which means road, path, or journey. A putnik is a traveler or passenger, and putovati is the act of traveling.

Why do we use the preposition na here instead of u for going 'to' the station?