Dramatic, yes, but also accurate.
I did not say that immigration laws are equivalent to or as egregious as the fugitive slave acts were, merely that they have much more in common with such unjust laws than they do with more fundamental laws with a clear basis in ethical principles that are (almost) universally supported.
There are definitely similarities though. The Fugitive Slave Acts were undeniably migration regulations. A slave plantation frankly meets all the criteria required for us to consider it to be a state, a particularly brutal one despite the lack of full sovereignty. The legal process by which a slave could move about legally required appealing to their owners rather than to bureaucrats purporting to serve a democratic state, but the principle is much the same. Slaves (and in some cases, free blacks) were forced to carry paperwork that was the equivalent of passports, visas, and work permits whenever they were allowed to venture off their master's property (usually to fetch him supplies or to be rented out as a laborer for someone else). The fugitive slave act punished not only the migrants (runaway slaves) but also the local citizens who chose to help them rather than acting as agents of the forces that imprisoned them. The same can be said of those who employ illegal immigrants, particularly if such draconian measures such as universal mandated use of E-Verify are passed.
Regulating migration has no place in "The Land of the Free." It has always been a hallmark of totalitarian regimes. The distinction between slave and free is blurry; most involuntary servitude in history has not been chattel slavery, but something more akin to serfdom. The defining factor of such a status is bondage not to a master, but to a territory which one is not allowed to leave without special permissions. There typically are also duties to serve in labor corvees and pay taxes, but those are secondary and usually justified as rent payments for what land the serf is permitted within the borders of the manor. A freely mobile worker would not have to accept such terms.
The bible actually describes the status of Israelites in Egypt not as chattel slavery, but as something very much like serfdom. When they escaped, they founded a society with no place for such restrictions on the movement of peoples. They were enjoyed to remember the plight of the alien and to love him. The Law of Moses not only welcomed immigrants, but explicitly granted the poor sojourners full and equal Gleaner's Rights (which many would call Welfare, although it does not require any State or government to administer) and even a portion of tithes during certain years. The prophets are pretty clear that Sodom and Gomorrah were destroyed primarily for how they treated poor immigrants.