I thought changing save device would help ,but I guess It is not ? There my knowledge of what can go wrong ends > there are some gamesaving problems like yours reported on PS3&PC and I've experienced some regresion in gameprogress (I've lost some singleplayer mileage , achivements and 1 MW car ) but dont recolect that anyone reported that It cannot save game on XBOX at all .
in rply to "Started the game as well ,it asked me again where do i want to save the game (that was strange) at that point..." > game always asks you after PRESS START to choose save/load device (if you got more than 1 storage device (internal,USB,Cloud) enabled) > it doesn't matter if save exist or not @ that device > if exist then it is loaded if not it is created/saved .
Can you try to go online , then drive a car for @ least 8 km to unlock 1st mod and then quit game > then boot again to see if game saved online progress > 1 unlocked mod online .
Keep thread bumped up and maybe someone will give you some usefull advice or moderators will proceed your problem to developer.
If I come across some similar problem I'll post you a link ...
BTW : do you have any other game to check if gamesaving works ok