Yes, it is possible for two corporations to have the same name but incorporated in different states. This assumes there are no trademark issues, unfair competition issues, etc. If same ownership, there should be no problem.
Yes it is possible for the same business to be incorporated as different corporations in different states - the states themselves will not check; however, that structure would generally not be preferred because it would create extra work and extra liabilities. For example, you might have the same shareholders, directors, and officers - but all playing duel roles to two separate corporations - and that could easily lead to duty of loyalty problems, such as if only one business checking account is used for two different corporations. Generally, better to incorporate in one state and register as a foreign corporation in the other state. Or alternatively, to have one of the corporations wholly own the other corporation, as that will help clean up duty of loyalty issues.