There is a "ground" but it's just a "common" or "return" or "reference" and not earth-ground. (A USB power bank might sometimes have an earth ground.) Your car is similar - The car body is considered "ground" and it's connected to the negative battery terminal but obviously the tires are insulated so there is no earth ground.
Scary and strange! Is that normal where you live? Is one of the power wires "neutral"? Here in the U.S., most power outlets have 2 "power connections" and a ground but some power plugs don't have the ground. There are 2 things that make that "safe". Usually the item is "double insulated" which means there is no ground but user is insulated from the voltage.
A desktop computer does normally have an earth ground through the power-cord connection (in most countries). But, that ground is completely isolated from the AC power so if that earth-ground is cut, it's still safe... as long as there is no fault.
There other safety feature is that one of the power-wires is "neutral" and connected to ground at the circuit-breaker box. The plug is polarized (one plug-blade is larger than the other) and on some older products the the neutral is connected to the chassis. This can be dangerous if the wires are reversed, or if a non-polarized extension cord is used, so they don't sell those things anymore. (If you get a shock you can turn the plug over to reverse the connections.)
In a lamp the screw-part of the bulb is connected to neutral so you have to stick your finger all the way in to get a shock. (If everything is wired right!)
Usually, a laptop's power supply is transformer-isolated so there is normally no direct-connection to the AC power or AC/earth ground. There is a often a ground on the AC side, but that's usually not connected to "ground" on the DC side. You can check that if you have an Ohmmeter and/or you might want to try a different power supply/charger. But, if you plug something into the laptop with it's own power supply, there can be a connection (hopefully to earth ground).
It won't be grounded unless it has it's own-separate ground. The "floating ground" can provide noise shielding if you have shielded cable or a metal box, etc. Normally, the Arduino (and you!) are safe, but if you are getting a shock I'd say neither you or the Arduino is safe! 
It's not dangerous to have a floating ground unless that floating-ground is "energized."
Normally, a "safety ground" works like this - If there is a fault that puts voltage on the ground (say, your computer case or a metal box containing your Arduino), that voltage is "shorted to ground" so you cant' get a shock if you touch the box and ground at the same time, and a circuit breaker will be tripped turning-off the power.