If the idea for the Institute had been the universities, and they were soliciting donors to fund it, and Pearson was one of multiple donors, sure that would qualify as a gift, but as the article is written, this was a very different beast.
UChicago didn't have to accept it. They chose to accept it, with the associated terms and conditions. The way the article reads, it looks like there's blame to go around, but possibly more to UChicago than Pearsons.
Normally when you make a tax-deductible charitable gift, the organization you donated to will send you a letter documenting your gift and stating that no goods or services were provided in exchange for the gift:
https://www.irs.gov/charities-non-profits/charitable-organiz...
https://www.irs.gov/pub/irs-pdf/p1771.pdf
The IRS might review that documentation as part of a tax audit. If it's inaccurate, it could conceivably affect the tax status of the organization that issued the acknowledgement letter.
The IRS can also revoke the 501(c)(3) designation of an organization if it concludes that the organization is inappropriately providing benefits to its donors. For example, it might require the organization to be reclassified under a different tax-exempt status where donations are not automatically treated as tax-deductible charitable contributions.
A subtle thing that might be bothering you is that people can make extremely specific charitable gifts to pursue extremely specific goals, and those can still be treated as tax-deductible. For example, you could donate to create a scholarship with very detailed eligibility requirements (often, donors to schools have created scholarships for students with very similar life circumstances to the donors' own). But, there is still a requirement that can't claim a deduction for the amount of a donation for which you personally receive goods or services with a market value as a quid pro quo, except to the extent the donation exceeds that market value.
I'm not a lawyer or accountant and this is not tax or legal advice.
When I joined the Bronx Zoo as a member, I gave them $200 and received some benefits in consideration. Those benefits are valued at $140, so I am able to deduct $60 from my income.
This is a first though, I didn't know you could take it back.