Would a modern processor/ALU process data bit or byte wise?
Modern processors have 32/64 bits integer and 32/64/80 bits floating-point registers and ALUs. With the addition of SIMD instructions, extra vectorized ALUs are available and can perform operation on up to 64 bytes or 32 shorts (16 bits) or 16 ints (32 bits) or 8 long ints (64 bits) simultaneously and similar figures in floating-point (hence 512 bits at a time - Intel AVX).

A core is a full processing unit packaged in the same physical (multi-)processor.

So in a way, you can say that an octo-core can process 4096 bits of data at a time. 



It is not fair to say that a processor ignores bytes, because the arithmetic operations need to know how many useful bits there are, to handle borrows, carries, overflows... Not to mention the floating-point data types, such that the value is split in several bit fields with different roles (sign, exponent, mantissa). In fact, a processor distinguishes dozens of data types of various lenghts.

It would be more true to say that processors don't known about bits because these are never handled individually, but as a minimum grouped in bytes. Even the very first microprocessor in history, the 4004, was handling nibbles (4 bits at a time).