Do laser Christmas lights have the same dangers as laser pointers aimed at planes?
A fine beam is needed do draw patterns. A regular led would require a moving focusing mirror, something expensive.

There is a non-standard warning: "Laser aperture, Max output &lt; 1 mW" near the leds:

  

As the leds are in the visible spectrum, this is equivalent in power to a Class 2 laser.


  A Class 2 laser is considered to be safe because the blink reflex
  (glare aversion response to bright lights) will limit the exposure to
  no more than 0.25 seconds. (Wikipedia).


This image shows the effect of a 5 mW laser light. You may use it for a 1 mW laser light by dividing the distances by 5:


Adapted from original on Wikipedia

or from the original site:


  
  A 0.99 mW Class 2 laser beam can temporarily flashblind a pilot or driver, causing afterimages, within 240 ft (73 m) of the laser.
  It can cause glare, blocking a pilot or driver's vision, within 1050 ft (320 m) of the laser.
  It can cause distraction, being brighter than surrounding lights, within 2 miles (3.2 km) of the laser.
  


The above distances are for a tightly focused beam, which is probably not the case of an inexpensive laser device. The effects of a more diverged beam are weaker.

More info: This hazard chart.

To sum up:


Most countries have regulations for dazzling a pilot, regardless of the mean and power used, it's the effect that is taken into account.
A class 2 laser led couldn't blind someone unless the person is at a few meters (15 m for temporary blindness and 3 m for eye damages, if the eye is forced to stay open).
As the class is not mentioned on the label, I would be careful when seeing this item.
There are actually 2 laser leds, a red and a green, is it 2x1 mW or 2x0.5 mW? In practical the lasers probably can't illuminate the same eye, due to the divergence of their beams.
To draw a pattern a laser light must move, so the time of illumination of a point is reduced when the device is working correctly.