No, all eight characters have a one-to-one correspondence with specific C code
that's like saying that because assembly and machine code have a one-to-one conversion they are the same.
Brainfuck "interpreters" exist, but those are just programs that convert brainfuck code into the equivalent C, which would then be passed through a C compiler to produce an executable binary.
that would be called a compiler, not an interpreter.
if you want a brainfuck interpreter here's one
Code:
void parse_brainfuck(char *code)
{
    unsigned int ptr=0;
    unsigned char memory[1000];

    while(*code!=NULL)
    {
        switch(*code)
        {
            case '>':
            ptr++;
            code++;
            break;

            case '<':
            ptr--;
            code++;
            break;

            case '+':
            memory[ptr]++;
            code++;
            break;

            case '-':
            memory[ptr]--;
            code++;
            break;

            case '.':
            putchar(memory[ptr]);
            code++;
            break;

            case ',':
            memory[ptr]=getch();
            code++;
            break;

            case '[':
            if(memory[ptr]==0)
            {
                while(*code!=']') code++;
                code++;
                break;
            }
            else
            {
                code++;
            }
            break;

            case ']':
            if(memory[ptr]!=0)
            {
                while(*code!='[') code--;
                break;
            }
            else
            {
                code++;
            }
            break;

            default:
            break;
        }
    }
}
you can also do one in assembly (and pretty much every other language), but i don't feel like typing that much