If dy/dx is not a fraction, then why am I allowed to multiply both sides of an equation by dx?

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The derivative (or in the case of integration, antiderivative) is always with respect to one or more variables; stating a problem as “dx = {non-differential function}” is meaningless unless you are applying the “nonstandard analysis” methodology in which the infinitesimal are represented by hyperreal numbers rather than placeholders for taking the limit of ratio of infinitesimals to get a real valued function or specific value at a point.

Thanks for bringing up infinitesimals. I remembered them but since I haven’t done anything with calculus in a long time, my desire to study and my understanding of hyperreal numbers is lost.