Is it Islamically permissible for a person to continue receiving financial aid from one European country after moving to work in another European country, and to call this "economic jihad"? And is it forbidden in Islam to cover up such an act?

The action taken by your friend is impermissible because it is a fraudulent scheme to unjustly acquire people's money. This is due to the fact that if the granting state knew he was working in another country, it would stop his grant. Additionally, he will be forced to lie to prove that he is still unemployed, and this contradicts Allah Almighty’s command to fulfill contracts, as He says: "O you who have believed, fulfill [all] contracts." This is a general command, and contracts must be fulfilled with anyone, whether Muslim or non-Muslim. The jurists have unanimously agreed that if a Muslim enters a land of war with a guarantee of safety or a truce, it is forbidden to harm any of their lives, property, or honor, for Muslims are bound by their conditions.

As for reporting him, we do not see that as necessary. Instead, it is sufficient to advise him and warn him against consuming unlawful wealth.