Certainly the current source in your first circuit is useless. Its terminals are the same node, so it does not interact with the rest of the circuit, just circulates current back to itself. You can delete it from your analysis — or split at that node into two independent circuits, one with the shorted current source and one with everything else — and get the same result.
However, this is not usually considered analogous to shorting out a voltage source. A shorted ideal voltage source is a contradiction, and a shorted non-ideal voltage source will either destroy itself, shut down, or reach a current limit which reduces its output voltage.
The analogue of this in a current source is an open circuit. If you disconnect one or both terminals of a current source (or break the circuit elsewhere), then it is impossible for the specified current to flow, so this is a contradiction with an ideal source, and a non-ideal source will reach a voltage limit (“compliance voltage”) and thereby fail to supply the specified current.
(And for completeness: the fourth case is a voltage source in an open circuit — which, like the shorted current source you drew, does nothing and can be deleted from the circuit.)