In a matching market each player gets at most one item. You are correct about that. The case in HW2 Q 3 is a matching market. There is also another assumption in the question - there is only one optimum (which is indeed, as you defined, a maximum weighted matching). It doesn't have to be like that of course in a general matching market. But in this question we prove that it leads, with the right prices, to a unique demand set for each agent.
Maybe it was confusing in this context, but in the previous answer I was talking about unit demand functions in general and the fact that adding an item with price 0 to a bundle of an agent can change the utility. You can think of many settings in which an agent can get a bundle of more than one item, even if she has a unit demand function. The fact that the function is unit demand doesn't restrict the number of item an agent can get. It depends on the mechanism and the restrictions of the market, and not the valuation function. You are right that the value for a bundle in a unit demand function is influenced only by one item, the most valuable item. When I said that adding an item with price 0 might change the utility I was talking about situation like the following: consider a unit demand function v over two items {a,b} which is defined as follows: v(a) = 1 , v(b) = 2. Consider a price vector: p(a) = 1, p(b) = 0. If the agent owns only {a} the utility is 0, if we add item b to her bundle we get: u({a,b}) = v({a,b}) - p({a}) - p({b}) = 2-1 -0 = 1 which is greater, even though we added an item of price 0.
I hope it is clearer now.
