To be fair, it was illegal, since he was already married in Kenya to someone else.
In support of your point — on the application dated 1959, Obama indicates that he is married.
The letters asking for financial support dated after the purported marriage to SAD and birth of Barack, Jr., omit to mention their existence, even though Obama rattles away gracelessly about all his specific costs and needs, from winter clothes, to rent and hamburgers.
On the other hand, the INS at some point was investigating Obama for bigamy (not sure if this was for SAD, or the other white woman he married). Immigration must have thought he had married one of them.