How can two countries of roughly the same size, population and GDP have such big differences in minimum wage? We could answer this question theoretically, or give a real-world example of two such countries.

Is there anything more interesting to say than: that's just something decided by the government? Could geographic location play a role? Is an equal GDP an indication of equal minimum wage, or isn't it?
The minimum wage is usually the outcome of a bargaining process between the government, businesses (via lobby), and the trade unions. As such, a key driver of the final outcome depends on the balance of power. In countries like France and Germany, with strong unions in the former, and wide collective bargaining on the latter, the minimum wage is relatively high, compared for instance with Poland. Meanwhile, in Poland the minimum wage is roughly two times that of Chile, both roughly equally sized in terms of GDP per capita. How this reflects differences in power is not easy to tell.

Needless is to say that the ideology of the government is also crucial in this respect. Left-wing governments are more keen on raising the minimum wage than right-wing ones.

Additionally, the structure of production and workers' skills varies among countries. This means that some countries could have a lot of workers around the minimum wage, giving more incentives to unions to fight for them, with greater opposition from business. Yet, businesses in a country with few people around the minimum wage are more willing to accept a minimum wage, because it affects less people.

Last but not least, what matters is the real minimum wage. Equally sized countries can have different price levels, because the basket of non-tradable goods might be very different (prices of tradable goods are supposed to equalise in the long run, according to the Law of One Price).