(Note: I am assuming that by China you are talking about the People's Republic of China (PRC). It's very different for the Republic of China (ROC).)
The word "recognize" means to treat something as if it exists or not. The precondition is that the thing exists. What's the point of "recognizing" or "not recognizing" something that doesn't exist?
China doesn't recognize dual citizenship.
You may be referring to Article 3 of the PRC Nationality Law, which says
The People's Republic of China does not recognize that Chinese
citizens possess dual nationality.
What does this mean? To "not recognize" that this person possesses dual nationality, is to consider this person to not have dual nationality, even if this person does. But by calling this person a "Chinese citizen", it is recognizing this person's Chinese nationality, so it is the other (non-Chinese) nationalities that are "not recognized". In other words, if a person has Chinese nationality and foreign nationality at the same time, China only recognizes this person's Chinese nationality, and does not recognize the person's foreign nationality. If there weren't any Chinese citizens possessing dual nationality, then there wouldn't be any point in "recognizing" or "not recognizing" Chinese citizens possessing dual nationality, would there?
If you think about it, this is not different from basically how every country in the world operates -- if you are a dual national, and you are in the country of one of your nationalities, then that country will treat you the same as other nationals of that country. You should not get special treatment just because you have other nationalities -- hence, your dual nationality has no "recognition".
What are the practical consequences today if a Chinese citizen also
acquires the citizenship of another country, but does not formally
renounce their Chinese citizenship?
Legally, if a person is already a Chinese citizen, and they then voluntarily acquires a foreign nationality, they automatically lose Chinese nationality by Article 9 of the PRC Nationality Law. Whether or not they take any action, or whether or not the government takes any action, does not, legally, matter. The person is supposed to henceforth present themselves as a non-Chinese-national (e.g. apply for a visa to visit China, etc.).
In particular, what happens upon entering/exiting China? Is the ban on
dual citizenship enforced, or is this situation mostly ignored?
First of all, since they have already lost Chinese nationality by operation of law, there is no "dual nationality" here. The person may potentially try to conceal the fact that they have lost Chinese nationality, and thus pretend they are still a Chinese citizen.
And yes, whether the person is pretending to be a Chinese citizen does come up upon entry/exit. For example, if a person flies directly from China to the country of one of his other nationalities, then the exit control will ask him for his document to enter that country. And this person's only documentation will also show him to be a national of that country, leading them to inquire how he obtained that nationality, leading them to find out that he acquired it voluntarily, and thus no longer has Chinese nationality. Or, if the person tries to renew his Chinese passport in the country of one of his nationalities, they will ask him for evidence of legal status in that country. And this person's only evidence will be evidence of nationality, leading again to the same inquiry.
Is there actually a ban on acquiring another citizenship
No country can "ban" acquiring other countries' nationalities. What would that even mean? For example, if smoking is banned, then if you smoke they can prosecute you. Does that mean they will prosecute you after you acquire another country's nationality? That would be absurd. Each country's nationality is solely determined by its own laws, so no other country can "ban" it.
In every country, there are circumstances when a person will, according to the law, have that country's nationality and another country's nationality at the same time.
or does "not recognize dual citizenship" simply mean that China is
going to ignore the other one, e.g. the person couldn't get consular
protection from the other country while in China?
As I've said above, "does not recognize Chinese citizens possess dual nationality" DOES mean that China will ignore the other one(s). However, that only applies to someone who really (legally) has dual nationality. If one has lost Chinese nationality automatically by voluntarily acquiring another nationality, then there is no Chinese nationality there to recognize in the first place.
You may wonder, who really has dual nationality of China and another country, according to the law (ignoring peculiarities with Hong Kong)? The dual nationality generally has to be gotten automatically at birth, because nationality at birth is involuntary. Here are some examples:
A child born in China to a Chinese-citizen parent and a foreign national parent. According to Article 4 of the PRC nationality law, the child has Chinese nationality automatically at birth. The child may also have the other parent's nationality automatically at birth through jus sanguinis.
A child born outside China to a Chinese-citizen parent who has not "settled abroad" and a foreign national parent. According to Article 5 of the PRC nationality law, the child has Chinese nationality automatically at birth. The child may also have the other parent's nationality automatically at birth through jus sanguinis. If the child is born in a jus soli country, the child will additionally have the nationality of that country automatically at birth.