One thing that is interesting here is the emphasis on corporations as "persons" versus the issue of governmental power overall. A case about a century ago suggests a possible difference. Ironically, given some opinions on the sentiments of the era, it noted:
In creating a corporation a state may withhold powers which may be exercised by and cannot be denied to an individual.
The dissent suggests, true or not, there are limits:
There is no magic in the fact of incorporation which will so transform the act of teaching the two races in the same school at the same time that such teaching can be deemed lawful when conducted by private individuals, but unlawful when conducted by the representatives of corporations.
Yes. The law in question, the one that limited corporations in particular, denied incorporation to those bodies that promoted the integration of the races. The Court later determined that when this inhibited sending a student to an incorporated religious school, it was unconstitutional in the case involved, an early security of the right to privacy:
But the injunctions here sought are not against the exercise of any proper power. Plaintiffs asked protection against arbitrary, unreasonable and unlawful interference with their patrons and the consequent destruction of their business and property.
Similarly, person or not, the state cannot selectively benefit certain types of corporations based on the religious messages of their ads. The government, pursuant to the 1st and 14th Amendments does not have the power to pass such laws. It might be determined that corporations are such that certain types of limits, including related to speech, are within the state's power. But, corporate personhood alone is not the test. Similarly, the limits to state power is in some fashion related to the effects on humans, as shown by the law banning incorporation of integrated schools.
And, as noted in the latter case:
Appellees are corporations, and therefore, it is said, they cannot claim for themselves the liberty which the Fourteenth Amendment guarantees. Accepted in the proper sense, this is true. But they have business and property for which they claim protection. These are threatened with destruction through the unwarranted compulsion which appellants are exercising over present and prospective patrons of their schools. And this court has gone very far to protect against loss threatened by such action.
[case citations omitted]
Someone is likely to have "standing" to hold the government account for passing a law that abridges the freedom of speech which is done by humans even here in some sense. People authorize Nike's speech, people listen to and learn something from it. People downloaded the video here. It can be argued that standing given to corporations here advance their interests as is done when some other advocacy corporation critics of the ruling often like is so given, including but not limited to media corporations. e.g., NAACP v. Patterson.
Even if the corporation is not a "person," it is possible the government does not have the power to inhibit the expression or other interest involved. If it does, corporate personhood alone isn't necessarily the deciding factor.