Products in quotient category

The issue that you mention is accurate; you can basically make all sorts of universal properties fail by making maps commute that didn't initially commute. An example that occurs 'in nature' is the homotopy category of topological spaces, which is obtained from the category of topological spaces by taking homotopy to be your equivalence relation. This homotopy category has bad categorical properties. Probably products are OK but general limits won't exist.

(In actuality, this is really a projection of the fact that $1$-categorical quotients ofte aren't quite the right thing to consider; indeed the $\infty$-categorical homotopy category has all limits and colimits.)

To find a proof-of-concept example where products fail to exist, let me try the following. (I'll have to ask you to fact-check what I'm saying though.) Consider a category $\mathcal{C}$ with four objects, $X$, $Y$, $0$, and $X \times Y$. Here $0$ acts as the zero object, and there are projection maps $X \times Y \to X$ and $X \times Y \to Y$ making the object $X \times Y$ into the categorical product of $X$ and $Y$. Now define $\sim$ by simply declaring the projection maps to be equivalent to the zero maps. Then I think that the product of $X$ and $Y$ in $\mathcal{C} / \sim$ will be $0$.