Many economic commenters, when broaching into the topic of induced demand, like to break into a car analogy. They assert more lanes actually cause more traffic rather than less.

Is there any proof of that assertion?

The few studies I've found on this suggest otherwise.

Haven't looked at any research data tbh, but guessing it just has to depend on more than that one factor? Location (mainly population density of the area), quality and availability of public transportation, parking availability in the (broader) area?

Lately, the city's been expanding some streets in the broader centre with extra lanes which I'm sure will provide some temp relief, but I'm thinking this will be an extra confirmation signal to more people looking to move into cars as soon as they can and ultimately lead to even worse jams and more parking issues.

No?

For example, Belgrade has regular traffic jams, growing population density, increasing parking issues, mostly no bike lanes, and no metro, just busses/trams which aren't really the best quality and often overcrowded (though currently cheap).

People like to move to personal vehicles as soon as they can afford.