Please tell me if it is possible to be "addicted" to drugs in general, and not one specific substance?

No: the term becomes meaningless when thought of in this way. Addiction is not just physical dependency, but a complex biopsychosocial response to life and its pressures. There are people who are risk-takers and will try everything, and potentially abuse everything, including non-drug issues like gambling, the internet, sex etc. This is more of a personality disorder issue than it is addiction.