I currently own two credit cards in different countries (my country of residence and my country where I am a citizen of) and they both have free travel insurance included through different insurance companies.
Could I run into any problems while trying to make a claim?
I know that I can't claim the same problem with both of them, but I also heard that the process could get long and complicated, because one company may try to get a part of the payout from the other company, is that true?
When you have multiple insurances covering the same thing, you can apply to any single of these insurances to get your damages covered, and they have to cover it in the same way as if they were your only insurers.
They may try to recover money from the other insurance, but (1) that doesn't affect you, and (2) the other insurance has no reason to give the first insurance any money, since there is no contract between them. And (3) a significant part of the cost for an insurance claim is handling the claim. You are not the only person with two insurances, and someone will claim at the other insurance, so it's cheapest for all insurances to just pay without causing each other extra work and cost.
It's just bad luck / good luck for the insurance if you ask the insurance to pay / ask another insurance to pay.
Typically, credit card travel insurance only covers travel that you pay for - completely - with that credit card.
As you can only 'pay completely' with one card, the problem doesn't really exist.
