You seem to be confusing concepts from partial and general equilibrium.
Supply and demand functions are usually mapped in such a way that quantity is on the $x$-axis and price is on the $y$-axis. Competitive equilibrium is where the supply and demand functions intersect.$^1$ It is called partial equilibrium because only the market for one good was analysed. Slopes play no role in determining whether a point is an equilibrium or not.
General equilibrum in pure exchange economies
General competitive equilibrium considers all goods. Lectures usually restrict themselves to two types of goods and two consumers, as that scenario is easier to understand but gives the same insights.
The most frequently used display method in pure exchange economies is the Edgeworth box. It uses the fact that the total amount of goods is given, an allocation just distributes these among the two consumers. Quantity of good $x$ is on the $x$-axis and quantity of good $y$ is on the $y$-axis. Distance from the lower left corner shows the first consumers basket of goods, distance from the upper right corner shows the second consumers basket of goods. Given utility functions one can analyse Pareto-optimality in this setting. An allocation is Pareto-optimal if the slopes of the indifference curves running through the allocation are identical.$^2$
Also given an initial allocation, which denotes the property of the consumers, one can derive the general competitive equilibrium. The equilibrium price ratio is where the price vector going through the initial bundle is tangential to an indifference curve of each consumer and the point of tangency is the same. The equilibrium good allocation is the point of tangency. As a result, here, the slopes of the indifference curves are identical, hence the equilibrium good allocation is Pareto-optimal.$^3$
Intersection plays no role in determening whether a point is an equilibrium or not. Since we draw the indifference curves that run through the same point of the Edgeworth box, they will intersect (run through the same point) for sure.
There are some exceptions when $p = 0$ but that is a special case.
There are some exceptions where the indifference curves are non-differentiable at the allocation and hence the slopes do not exist. The allocation may still be Pareto-efficient. In case the allocation is on the side of the Edgeworth box there are further exceptions.
Again, this is only true if the allocation is in the interior of the Edgeworth box and both indifference curves are differentiable.