User power supply current limit to avoid current limiting resistors for LED
No, it is not safe to assume that the current will be divided equally between LEDs in parallel, as there will invariably be differences between the LED dies.  They might be closer if from the same batch, but it is still a bad idea.  What would be closer to working would be connecting them in series, using a higher supply voltage.  Some cheap flashlights actually work that way, and depend on what could be crudely modeled as the Thevenin resistance of the batteries to limit current.
You cannot trust a supply to not source more current than it is rated for, unless it is explicitly rated as having a controlled current limit.  If it does, there will be a tolerance specification on the parameter which will tell you how much higher or lower it could be while remaining within specification.  In contrast, if it is merely something such as a transformer core in a linear supply starting to saturate, that would be a fairly imprecise and unreliable limit unless that core were carefully manufactured and tested for saturation current.
If the current limit is an intentional feature, then it may be designed to operate in current-limited regime - check the data sheet to be sure.  If the current limit is a design artifact, then it may not be appropriate to operate there for an extended period of time (could overheat, etc).


In short, what you want to do is not a good idea in this form.  However, connecting your LEDs in series and using some sort of active current regulation would be a good idea - non-trivial to get right, but potentially much more energy efficient than using resistors.  I believe if you do some searching you will find designs for things like LED bicycle safety lights which do current regulation the right way, in order to maximize battery life.