Earlier quoted context omitted.
30k is gross and 13k is net income, those are very different things. You also only looked at the poverty line for singles, not for couples (or single parent) with children. But you are in good company in this thread as nearly nobody takes children into account. Raising children on the poverty line is not the right way.
The poverty line is defined as 60% of median income, not by needs. In the UK there is also the phrase “living wage” which seems to be about needs rather than relative wealth, but I don’t know enough German to get past the opening paragraph of news (I have the German vocabulary of an 8-year-old, which is somehow all you seem to need for CEFR B1) so if an equivalent exists here, I don’t know it yet. Fair point about gr…
Poverty often means that you cannot afford a new washing machine when one breaks. Or not it being able to send your children to school trips.
As a German, I can tell you: 30k€ is not enough to raise a child here, even if both parents are working because at least one cannot work full-time.