Using a power bank to run nebulizer
Summary:


Will probably operate OK on a 5V "USB powerbank  


Given the "USB" connection is very informal you need to make your own decisions on safety (for the equipment) of doing this. It's probably OK. 

If it does then operating time seems likely to be in the order of 1 hour per 1500 mAh of powerbank 3.6V LiIon cell capacity.  


This is based on the run time spec you provided and may differ with brand and model



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Added:  Much material below is redundant due to an initially poor specification.
It seems likely but not certain that the unit is "designed" to accept 5V input from a "USB 5V supply" using a non standard cable. 




  I have a portable nebulizer that can be powered by either an external 3V power supply or 2 x AA batteries AA batteries. Expected run time (if the same as a prior model) is about 12-15 minutes
  on either AA Alkaline or AA rechargeable batteries.
  
  Would it be possible to safely use a power bank to operate (not charge) the nebulizer? 


Despite your original claim that it used a USB supply it appears that it doesn't. USB supplies are 5V and use one of the USB standardised plugs.  USB supply NEVER uses a round plug.




  The description photos shows a non normal USB cable and states USB



There are no "non normal" USB cables - only USB cables and others, some of which have one or other connector that matches those on USB cables. This is a dangerous practice usually done to save cost or to "update" an old product. 
In the link that you provided there was no mention of USB.
They DO show a symbol similar to the std USB logo and it will plug into a USB socket. IF it works safely and properly on a 5V USB outlet it will also (probably) work OK on a USB powerbank. 




If your nebuliser supply provides 3V then it is NOT a standard USB supply.

Please provide 


Q1: Brand and model of nebuliser and 
Q2: If shown - Specifications shown on the "USB" power supply.
If not shown on supply, say so.
Q3: Measure supply output voltage with a voltmeter. This may be greater than 3V unloaded. 


IF the nebuliser accepts standard 5V USB power input then a power bank providing 5V USB power will* operate it. 

If the nebuliser supply is nominally 3V rated then a 5V supply may damage it.
It is possible to make a 5V to 3V converter.
Details can be provided in due course if needed. 



The following was written assuming a 5V USB supply was used. It still applies if a 5V to 3V converter is used.

An AA Alkaline cell provides around 3000 mAh of capacity.
Two x AA cells provide about 3.2V when charged and about 2.2V when "flat".
Mean current for 15 minutes operation is about 750 mA - or probably somewhat less. That is a heavy load for most AA Alkaline cells.

If you assume that the nebuliser uses and internal linear regulator and watts the excess voltage above about 3V then it will probably draw ABOUT 750 mA from a USB input.
The power output of the power bank is then about P = V x I = 5V x 0.75 =~ 3.75  watts.
Power from the powerbank battery is probably ABOUT 5 Watts.
Current from the powerbank battery is then about I=P/V = 5/3.6 =~ 1.4A.  

So you'd hope that a powerbank would operate the nebuliser for about 1 hour per 1400 mAh of battery capacity.  So a 3000 mAh powerbank for about 3000/1400 = 2 hours +.
A 5000 mAh powerbank for about 5000/1400 =~ 3.5 hours +.

Actual times will depend on the assumptions above, but that gives some idea.




re "will operate it" -> If the nebuliser accepts a USB mini or micro plug to power it then it is almost certain that it will use any 5V USB supply capable of supply say 1A or more. In a very very very few cases people may use non standard arrangement with a USB plug but this is (fortunately) exceedingly rare.