Taḥrīr means removing ownership of the person in the historical legal context of emancipation. Ṭalāq concerns dissolution of the marital entitlement described in this juristic vocabulary as milk al-mutʿah.
The common element is termination of a legal entitlement, but the two acts are not identical. In the analysis under discussion, release from ownership is treated as a related mere means for the derivative marital effect, allowing figurative transfer in a specified direction.
Shared termination does not authorize swapping all the forms of divorce and emancipation in either direction.