By the sounds of it the EU ETS money does not pass through EU coffers. Member states implement the EU ETS nationally and revenues can flow directly into member state budgets as long as the "equivalent in financial value" of a determined share of the revenues is used as stipulated by the scheme:
Article 10(1) of Directive 2003/87/EC states that:
[...] Member States shall auction all allowances [...]
Member States shall determine the use of revenues generated from the
auctioning of allowances. At least 50 % of the revenues generated from
the auctioning of allowances […] or the equivalent in financial value
of these revenues [...] should be used [...] to reduce greenhouse gas
Further, the 2015 EU ETS Handbook (p.36) states:
It should be noted that not all auction revenues go to Member States.
Up to 300 million allowances from the New Entrant Reserve (NER), the
so-called NER300, are sold on by the European Investment Bank. The
revenue of these allowances are to establish a demonstration programme
comprising the best possible Carbon Capture and Storage and Renewable
Energy Supply projects, involving all Member States (see NER300 fund
for demonstration projects).
This report by Ecologic Institute discusses the details of member state reporting their use of EU ETS revenues and proposes improvements:
[Member states] can decide whether they allocate the revenues from
auctioning of allowances directly to a fund or support programme, a
process known as earmarking, or count the auctioning revenues as an
additional income stream to the state budget.
There have been discussions to transfer national ETS auction receipts to the EU central budget, see e.g. here (2018).