Is computer game addiction similar to alcohol addiction, drug addiction, or a gambling addiction?

Yes. Anything that a number of people do compulsively to the detriment of their health or function, qualifies as an addiction.
Addictions differ. Addictions to chemical substances (eg, alcohol, heroin, nicotine) reflect the original meaning of the word addiction, which includes biological drug tolerance and withdrawal symptoms as well as compulsive behavior to maintain the addiction. Behavioral addictions (eg, gambling, sex, internet, food) are similar in some respects and different in others. All can be serious problems.