Multiplication or division by a power of 2 can be done by bit shift assuming binary numbers represent base-2 numbers; i.e. not a beta-expansion https://en.wikipedia.org/wiki/Non-integer_representation where binary numbers are base-1.5 or base-sqrt(2) or base-(pi-2) or whatever (in which case multiplication or division by powers of 1.5 or sqrt(2) or (pi-2) could be done via bit shift).

But when multiplying two arbitrary floating point numbers, your typical case is multiplying base-2 numbers not powers of 2, like 1.01110110 by 1.10010101, which requires a real multiplier.

General floating point addition, multiplication and division thus require fixed-point adders, multipliers and dividers on the significands.

Fixed point might usually put the "binary point" in between bits, but when doing a multiply between two of them you still have to do at least an integer multiply before the bit shift. Ditto division.