According to the European e-Justice Portal [1], the right to legal aid is contained in art 6(3)(c) of the ECHR and art 47 of the CFR. However, the first right only applies in criminal cases and the second right is limited 'in so far as such aid is necessary to ensure effective access to justice.' Apparently, 'the broad objective in some States seems to be to make legal services and access to justice generally available, whereas in others, legal aid can be available only to the very poorest.'
Based on that summary, it seems like the position in most of the EU is probably much the same as the common law world (US/UK/CA/AU/NZ): you'll probably get legal aid if you're destitute and facing potential imprisonment, but you're unlikely to get legal aid to defend a copyright infringement case, especially if your net assets are non-zero and you can easily make the case go away by just removing some web links. That may mean that you are thereby deprived of your rights, if your legal argument is correct, but resource-starved legal aid bodies will prioritise cases that involve a smaller chance of someone being wrongfully convicted.
[1] https://e-justice.europa.eu/content_legal_aid-55-en.do