Infrared Mass Surveillance + Facial Recognition
China already has mass surveillance that makes it an authoritarian dream. A species powerful enough to shapeshift would pose a huge threat to national security - so surveillance seems like the logical, and politically appropriate, solution.
I recognize that the question states
we have zero tolerance toward ... extreme forms of discrimination
But this feels contradictory to your setting - a modern country in which Muslim minorities are detained in concentration camps to "keep the state safe". I don't want to make this answer political, but making shape-shifting illegal seems very likely, especially in this particular country. If humans that are different are subject to intense surveillance and discrimination, non-humans will special powers should fare far worse. For plot purposes, even if your modern China is non-discriminatory, surveillance of magical creatures is still integral to security.
For inanimate forms:
Assuming that shapeshifters taking the form of inanimate objects are still alive, they will probably still emit some heat. You can adapt mass surveillance systems to see infrared light. A sufficiently advanced AI could monitor each video feed, picking out objects of interest - similarly to how people identify the shapes of suspicious devices at an airport luggage checkpoint.
When inanimate objects exhibit an unusual glow in the infrared, they are targeted by drones or police officers. If the shapeshifters don't transform, they will be shocked by specially designed tasers to force a transformation - and then detained and relocated.
For human forms:
Piggybacking off the mass-surveillance idea, facial recognition techniques would keep a catalog of all shapeshifters and non-shapeshifters. If a face shows up in two locations at once, both people are arrested and interrogated for impersonation. If an undocumented face shows up, that person is similarly detained.
The benefit of these systems is that they are non-invasive (at least, they don't require physical contact), they work at a distance, and they can monitor almost any public space.
Gap in surveillance: Animals
Any shapeshifter that imitates a non-human animal (i.e. a rat or pigeon) will not be picked up by the facial recognition system, and will glow in the infrared like a real animal. This can be used as an interesting plot point / evasion tactic. Maybe the city catches on, and exterminates all birds / rats as a result.