How can I do two-factor authentication? I can't find my country (Serbia +381) and I can't verify my identity...

Serbia doesn't appear to be one of the countries were Two-Factor Authentication is available:

Availability of two-factor authentication for Apple ID - Apple Support

I am having the same problem, except I am in South Korea (+82).

Why not where I am living? I am going to be forced to have to make my family in the US be my "verification"...? That doesn't sound much safer to me...

Nobody here can speak to why some Apple services are not available in any particular country or region. Nor does anybody here have any knowledge about when or if a service may be offered in a currently unsupported region.

Bottom line though is, not all Apple support features are available everywhere. And if the service is not available in South Korea then you obviously cannot use it in South Korea with a local number.

You can still use 2 Step Verification in South Korea: Two-step verification for Apple ID - Apple Support