There is no straight answer to this question. Basically, every hotel/brand has its own policy when it comes to amenities. Then, inside the hotel, every housekeeper will eventually manage its inventory and use it differently.
Then, from my own experience, what I have seen :
1/ You'll most of the time get new ones if you have started using the ones available initially. This is especially true for shampoos and shower gel. For soap, it is more random and depends on the soap size.
2/ When amenities are high grades (like famous beauty brands), you less frequently get new ones. This is because their costs is obviously higher. We should take here into consideration the fact that in some rooms/suites, the value of amenities can sometimes cross 50 €...
3/ What I have seen is that every time you ask for new amenities, you get those. Never got a negative answer on this.
4/ If you have left the items unused, then you'll rarely get additional ones, especially if they haven't been moved.
And yes, hotels expect that you'll take the items with you, especially if those are high grade. This is what most of the guests are doing. Few reasons for that :
1/ People tend to think that the quality of hotel amenities is usually better than what they can buy at the supermarket. This is sometimes right, sometimes wrong (especially for hair conditioner).
2/ People see this as a kind of gift/souvenir that they are happy to bring back with them.
3/ Hotels market those in a way that you'll take them. This is a way for them to introduce themselves in your bathroom and make sure you remember about the good time you had at their hotel. It helps building branding. This is also true for beauty product brands working with hotels.