It is very close to being a TM, but there are differences, because of the way you describe the DFA in Brainfuck in contrast to a TM. Also, TMs are not entirely

It is very close to being a TM, but there are differences, because of the way you describe the DFA in Brainfuck in contrast to a TM. Also, TMs are not entirely formal in the sense that there exists multiple TMs which all are equivalent - but there is no correct implementation. So when people say TM, or even BrainFuck, then it is not entirely clear what machine they mean.