The European Union (E.U.) is not a country. Each E.U. member has its own legal system. There is not a single "EU law" or "legal system used by the EU".
Since Brexit, almost all of the members of the E.U. have civil law legal systems, except Ireland, although some Scandinavian countries are less true to that model than other E.U. countries.
Case law exists in both civil law legal systems and in common law legal system. But, the way precedents are used is different, and in a civil law legal system, in theory at least, all case law ultimately traces back to some legislative enactment, which is not true in a common law legal system.
Also, keep in mind that the European Court of Human Rights and the European Convention on Human Rights are not part of the E.U., they are part of the Council of Europe, which is a separate international organization from the E.U.
For the most part, E.U. law is part of the law of treaties and administrative law. Mostly, like other treaties, E.U. legal provisions are directed at getting member countries to act in particular ways, not at directly regulating the conduct of private individuals and entities.
Under primary law, the EU has only limited powers of enforcement, as EU law is usually enforced by the Member States. Furthermore, Article 291(1) TFEU adds that ‘Member States shall adopt all measures of national law necessary to implement legally binding Union acts’. Where uniform conditions for implementing legally binding Union acts are needed, the Commission exercises its implementing powers (Article 291(2) TFEU).
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There is no E.U. criminal law. There is no E.U. law of contracts or torts. There is no E.U. law of real property or divorce or inheritance. The E.U. provides guidelines for the tax laws of member countries but has little direct taxation and tax collection of its own. The E.U. doesn't have a military. Thus, the scope of E.U. law scope does not extend to the core subjects addressed by civil codes in civil law legal system and by the common law, in common law legal systems.
But to the extent that E.U. specific treaties and administrative law have a flavor, for example, in their European court procedures, it is closer to the civil law system of the lion's share of its members, than it is the the common law legal system.