If we say that Umrah is obligatory, then must a Muslim perform a new Umrah, intending it as an obligation, or is the previous Umrah, which he intended as supererogatory, sufficient for him?

Umrah is obligatory once in a lifetime, like Hajj. And whoever performs Umrah after reaching puberty has fulfilled their obligation and discharged their duty, even if they believe it to be a Sunnah rather than obligatory. For Umrah does not require the intention of obligation or specific designation; rather, the intention to enter into the ritual suffices. Thus, for Hajj, Umrah, and ritual purification (Tahara), the intention of performing the act is sufficient, without needing the intention of obligation or specific designation.