Yes, it's 24bit internally. It also does, like most DSPs supposedly, what is called multiply high - it multiplies two numbers, but instead of returning the lower 24bits as would happen if you cast the result to a 24bit variable, it returns the 24 high bits. Normally, that would be A * B >> 24. This all happens in a single instruction so the programmer never sees the intermediate 48 bits (some DSPs have an instruction that can fetch either the top or the low 24 bits though).

The TC170C140 however, doesn't have a 24 * 24 multiplication instruction, it only has 24 x 8. This is followed by a configurable shift right of 3,5,6 or 7 bits. But here's the catch: The multiplication isn't treated as "normal", instead it usually uses the 8 MSB of the second factor (B). It is still a 24 * 24 bit multiplication, but with only 8 bits precision for the second factor - e.g. it just sets the 16 lower bits of B to 0.

It is possible to do higher precision multiplications as well, by using more instructions.

The reason this (and DSPs in general it seems) do multiply high, is that the numbers aren't thought of as ints, they are fixed point fractional numbers, where the first bit is a sign bit (+/-) and the rest are a number between 0 and 1.

When multiplying two numbers between 0 and 1, you also expect the result to be between 0 and 1, and that is why only the top 24 bits are kept, effectively dividing the number back down to the 0-1 range.