Most laws and legals systems start with the unstated presupposition that they apply to Homo Sapiens living within the jurisdictions. Computers, animals, works of art etc. are treated as property, and are not considered to have rights or agency in the same sense that human beings do. This is true even if we know through observation and experience that dogs, great apes and other animals do have rudimentary forms of agency.
Now this is currently accepted since while dogs or advanced neural network computer systems are "smart", they do not have the full spectrum of intelligence needed to be able to enjoy all the rights or exercise the duties of citizens who do have rights under the law. This also applies to children and people incapacitated because of mental illness or developmental issues.
The problem is since much of this is "unspoken" and there are few qualitative definitions of intelligence or what it actually means to be human. This may become a real life issue in a few decades when true AI appears, and likely if humans ever start tinkering with their genomes to create post-human creatures.
So long as the humans retain the actual levers of power, they can legislate whatever rights or restrictions they want on AI or post-humans, and if they are in a minority, there is little they can legally do about it. Should they try to do something about it through violent action, they will be at the wrong end of legalized use of force and suppression to end the insurrection (since that is what it will be considered).
So from a world building POV, the introduction of this sort of genetic engineering would most likely be met with revulsion and legal restrictions. People who are genetically modified beyond some sort of accepted "medical" reasons (defined by law) could find themselves outlawed, deprived of property and effectively becoming wards of the State or property of whoever is deemed to "own" them. And if they use their "animal" natures to fight, then they may be subjected to hunting by the authorities, or worse yet, have bounties placed upon them so ordinary civilians can hunt them for sport.
The idea of a "civil rights" movement for modified humans might be delayed or even aborted should the modifications be extreme enough to make it difficult to identify modified people as "human". Indeed, other genetically modified humans who look human and AI might make this revulsion work to their advantage, getting humans aboard giving human looking modified humans or AI safely encased in a computer rights, while excluding non human looking modified humans. Humans themselves might exploit these sorts of divides to prevent modified beings capable of uniting and combining their abilities to upend the existing Homo Sapiens order.
