Do we need to check for mantissa overflow in floating point multiplication? We do check for the mantisas overflow in floating point addition

e.g.

If we are adding $8.02 \times 10^3 + 9.01 \times 10^3 =17.03 \times 10^3$ i.e we get an overflow, so we shift the number right and increase the value of exponent.

But does it occurs during floating point multiplication?

According to my logic, it should occur. because $9.99\times9.99=99.80$ which is a mantissa overflow, but that's not the case.

I have referred to Morris Mano's books and William Wtallings Computer Organization and Architecture book but none of those books mentioned about floating-point multiplication mantissa overflow.

So I feel like I am wrong.

Please tell me where I am wrong?
When we multiply, the answer is stored in 2 registers. When we multiply 2 numbers the results length does not exceed  ($n_1+n_2$) where $n_1$ is length of 1st number and $n_2$ is length of second number. So there is no chance of overflow here as the length of 2 registers is greater than or equal to $n_1+n_2$.