Haskell, 35 41 bytes

a#b|all(`elem`[0..99])[a,b]=mod(a*b)$1+99

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Example usage: 3#4 returns 12, while fake numbers like 3.5#4, 300#4 and (-3)#4 all yield the error non-exhaustive pattern in function #. Non-fake inputs whose product would be greater than 99 result in the product modulus 100.

Edit: Thanks to Jonathan Allan for pointing out that I initially missed a part of the challenge.