I currently live in Italy with my wife who is an Italian and I have Italian residence permit card, can I apply for a Norweigian reisidence permit if my mother lives in Norway and is it possible to hold the two residence permits at the same time?
Common sense suggests it's difficult to fully “reside“ in two countries at once. Either you care for your mother (the main way to qualify for a family member permit for extended family) and being there every other week isn't enough or you actually live with your wife. Splitting your time equally between both countries might just meet the standard for residence in each of them but it feels like you would be awfully close to lying about your intents when applying for a permit.
In any case, you can certainly apply for a new residence card elsewhere, there is no rule against that (that's also what you would be doing if you were to move permanently to Norway and never come back to Italy) and, to my knowledge, no system to exchange information between countries about those residence cards.
What could happen is that, come the time to renew your Italian residence card, it would be refused because you don't meet the definition of a resident anymore or are found to have effectively left the country. There are several reasons why it might still work in this case but that's what you should worry about. And vice versa for Norway.