When Jews pray the psalms and when Catholics pray the psalms, while they may praying the same text, they are often praying for diametrically opposed outcomes, especially having to do with the exaltation of Israel/Sion. In this sense we do not “share” even the Old Testament.
It is for this reason that Judaism unlike other non-Christian religions stands in unique opposition to the Catholic Church. It’s not just that they don’t believe in Jesus, but they received an identity from God that is itself a sign pointing toward something they reject. Christianity is not just an ideological opponent, its very existence creates for them a perpetual identity crisis at the most foundational level of their Jewishness.