Dimensionality of the target for DQN agent training
I am of the opinion that this architecture is only one among others that can solve the same problem (for example one may have only 2 outputs one for the chosen action and one for the $Q$ value of that action, but I will not elaborate further on this).
What this architecture does is to output the whole function $Q(a)$, that is the $Q$ value as a function of action $a$. So each output node represents the $Q$ value for a certain action $a$ corresponding to that node (so node 1 corresponds to the $Q$ value for action $a_1$, node 2 to $Q$ value for action $a_2$ and so on..)
BUT this is NOT a vector output, in the usual sense of the term. This is a functional output that represents the whole function $Q(a)$ for each action $a$.
As far as the learning/decision rule is concerned things are as usual. Hope this is clear.