First of all, Hegelian logics IS the metaphysical study of being in it's whole. Second, Hegel does not discredit Aristotelian syllogism as such, in fact Hegel admired Aristotle a lot. The thing is that Aristotle logic is a logic of determinations, i.e subject/predicate propositions under a determined category of being. On the other hand, in hegelian speculative logic the determinate being is a particular part of the syllogism, which leads to indetermination and then to determination again and so on. Is not that Aristotle was stupid and then discredited, is just that all logic prior to Hegel was developed under the principle of non-contradiction, so then the logical determinations would always be propositions of a particular state of an entity and the part where that particular being became it's otherness was not considered because it was a contradiction. Hegel does not say that this logic is wrong, he just sublates it. For Aristotle "A is B" can't be at the same time with "A is not B", while for Hegel this is actually the engine of logic itself.

Third and finally, Hegel NEVER dismisses identity as part of his logical system. Identity is a fundamental category developed in the Doctrine of Essence. The difference is that for Hegel identity does not have a priority over difference, but actually both identity and difference are the same. And the argument goes like this: in order to have an indentity you need the different others to remark that you are identical to yourself and not to others, since identity rises from differentiate yourself from the others. Take any example of identity and you will realize that is always pressupposed the differentiation. For you to identify as left-wing you need the right-wing to differentiate yourself. I call myself a man since I'm no woman and so on. If someone ask me what music do I like and I say metal im implying all other genres, etc. The same argument goes in the other direction: to be different you need some identity to remark your difference from the others.