Short answer: No European language is closely related to Mongolian. They belong to entirely different language families with no demonstrated genetic connection.
Mongolian → Mongolic family (core: Mongolian proper, Buryat, Oirat/Kalmyk, Dagur, Monguor, etc.). Part of the disputed Altaic macrofamily hypothesis (with Turkic, Tungusic, sometimes Koreanic, Japonic)—rejected by most mainstream historical linguists due to insufficient evidence for common descent vs. areal diffusion.
European languages → Overwhelmingly Indo-European (Germanic, Romance, Slavic, Celtic, Baltic, Hellenic, Albanian, Armenian, Indo-Iranian branches). Exceptions: Uralic (Finnish, Estonian, Hungarian, Sámi), Turkic (Turkish, Tatar, Bashkir, Chuvash—in European Russia/Turkey), Basque (isolate), North Caucasian families, Maltese (Semitic).
Systematic sound correspondences in basic vocabulary (numbers, kinship, body parts, pronouns), shared irregular morphology, reconstructable proto-forms. None exist between Mongolic and any European family.
Other Mongolic languages: Buryat (Russia), Oirat/Kalmyk (Russia/China/Mongolia), Dagur (China), Monguor/Tu (China), Kangjia, Dongxiang, Bonan, Shira Yugur. These share a common ancestor ~1,000-1,500 years ago (Proto-Mongolic).
Mongolian stands alone in Europe's linguistic neighborhood. Its nearest genealogical cousins are in Siberia, Inner Mongolia, and Qinghai—not Budapest, Helsinki, or Istanbul.