In brief: No — and Hegel used the expression polemically himself. „Philosophy of identity“ was the name for Schelling’s absolute, in which all differences vanish. Hegel’s whole procedure is the attempt to achieve the opposite: a unity that does not efface difference but holds it out.
Where the expression comes from
Around 1801, philosophy of identity meant Schelling’s system of the absolute indifference of nature and spirit. In his first publication — The Difference Between Fichte’s and Schelling’s System of Philosophy (1801) — Hegel still took its side. Six years later he broke with it, and in such a way that the friendship broke with it.
for the night […] in which, as one is wont to say, all cows are black — [this] is the naivety of emptiness in cognition.
Phenomenology of Spirit, Preface
The reproach is not that Schelling pitches unity too high, but that a unity from which nothing follows is not cognition.
What Hegel says instead
The famous formula stands in the Science of Logic, at the passage on the beginning. Analysing the beginning yields:
the concept of the unity of being and non-being — or, in a more reflected form, of the unity of being-differentiated and being-undifferentiated — or the identity of identity and non-identity.
Science of Logic I
This is no play on words but an instruction: the unity in question contains difference as its moment. Whoever thinks identity alone thinks nothing; whoever thinks difference alone gets no further than an enumeration.
Accordingly Hegel treats identity in the logic of essence not as a supreme principle but as the poorest of the determinations of reflection. The proposition „A = A“ says nothing; as soon as one takes it seriously and asks what A is, one is at difference.
The test in the philosophy of right
The standard by which Hegel sees Plato’s republic fail is precisely this one. § 185 Remark says that the ancient states lacked
the truly infinite power which lies solely in that unity which lets the opposition of reason go forth into its whole strength and has overpowered it, so preserving itself in it and holding it together within itself.
Groundlines, § 185 Remark
„Letting it go forth into its whole strength“ — that is the opposite of a philosophy of identity. And § 260 correspondingly requires that the principle of subjectivity be allowed „to complete itself into the self-subsistent extreme of personal particularity“. Civil society, with its competition, its inequality and its poverty, is not harmonized in § 182–§ 256 but presented in its dividedness.
What is nevertheless right about the reproach
Two things. First: Hegel claims that the oppositions are resolved in a higher unity — and this claim cannot be proved, only shown in the particular case. Whoever disputes that it succeeds at a given point has the objection that matters. Second: there are places where the unity is asserted rather than derived — the derivation of the monarch in § 279 is the best-known example.
The reproach therefore does not hit the programme, but possibly its execution. That is a distinction worth making.
Sources. The Difference Between Fichte’s and Schelling’s System of Philosophy (1801). – Phenomenology of Spirit, Preface. – Science of Logic I, „With what must the beginning of science be made?“; Logic II, First Section (identity, difference, contradiction); Encyclopaedia (1830) §§ 115–120. – Groundlines § 182–§ 256, § 185 Remark, § 260, § 279. – Literature. Dieter Henrich, Hegel im Kontext, Frankfurt a. M. 1971; Klaus Düsing, Das Problem der Subjektivität in Hegels Logik, Bonn 1976. – In the glossary: absolute idealism, contradiction, the speculative.