No, a "standard multiple regression" is not appropriate here, assuming by this you mean a regression with a single continuous variable as the response. Regression of this sort can only makes any sense if the different levels of the response can be seen as different values on a continuous variable. There is no way this can be the case with Subject. Even if it were you would have a lot of problems with dealing with the usual assumptions in fitting such a model.
I don't know how your stats package let you do this - probably it converted Subject into a continuous variable based on its internal coding eg 1=Maths, 2=business, etc probably in alphabetical order - but it will certainly have given meaningless results.
A chi-square test would tell you if there is a relationship between the variables, but if you want to understand whether Big 5 and Learning style are related specifically to Subject, you will probably be best off with a multinomial regression.