To put it short: there are two different kinds of NPOs, first "regular" e.V. and then those e.V. that fulfill exclusively "aims for the common good" ("gemeinnützige Zwecke", the full list is in §52 AO [1]) - they carry a special benefit: donations can be deducted from your income for tax purposes.
The CCC is a non-profit organization, but since it (among other things) engages in taking political stances while at the same time not being a political party, it is not seen as a "gemeinnützig" organization - a fate that hit quite a few organizations in the last years [2] or is looming over their head [3].
[1] https://www.gesetze-im-internet.de/ao_1977/__52.html
[2] https://www.campact.de/ueber-campact/der-verein/
[3] https://www.mdr.de/nachrichten/deutschland/politik/brandbrie...