I’ll start with various Oaths of Office:
The President of the United States of America: “I do solemnly swear (or affirm) that I will faithfully execute the Office of the President of the Untied States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States.
Arizona Governor: I (name) do solemnly swear (or affirm) that I will support the Constitution of the United States and the Constitution and laws of the State of Arizona, that I will bear true faith and allegiance to the same and defend them against all enemies, foreign and domestic, and that the I will faithfully and impartially discharge the duties of the office of Governor according to the best of my ability, so help me God (or so I do affirm).
New Mexico Governor: I do solemnly swear (or affirm) that I will support he Constitution of the United States and the Constitution and laws of this State, and that I will faithfully and impartially discharge the duties of my office to the best of my ability so help me God.
Texas: IN THE NAME AND BY THE AUTHORITY OF THE STATE OF TEXAS, I (name) do solemnly swear (or affirm) that I will faithfully execute the duties of Governor of the State of Texas, and will to the best of my ability preserve, protect and defend the Constitution and laws of the United States and of this State, so help me God.
In all four oaths, it is either implied through the duties of the Executive branch from the US Constitution or explicitly stated that the President and Governors will enforce the laws of the US and of their own States. The term “catch and release” is not codified anywhere in the US statutes. In fact, it is a phrase that has been cobbled together thru a collection of policies, court precedents, executive orders and some Federal statues. There is no LAW regarding “catch and release”. The practice is entirely based on these cobbling factors. It is a nifty, but non-legislated, trick. If there can be such a thing, it is neither legal nor illegal and is solely based on what the current President wants and does through Executive Order. Compare Trump’s EO’s to Biden’s EO’s.
However, legal or not, it begs the question of US sovereignty and the protection of US and State borders and of those who legally reside therein. To quote Abraham Lincoln: “We are now engaged in a great Civil War…”