The EU has a lot of software projects and a lot of them are open source (EUPL licensed, which is compatible with most other FOSS licenses, it's sort of a EU-compatible GPL). The problem is that most of it is self-dogfooding software, ie, software that meets the needs of the EU governance, which is not necessarily something that is a need in a lot of other institutions like corporations.

You're, however, free to release software under EUPL to indicate you support the EU in the free software effort.