Assignment of an invention involves conveying all substantial rights in the invention–the exclusive rights to make, to use, and to sell. These substantial rights can be conveyed expressly by transferring title (that is, the right of ownership) or implicitly by licensing exclusively all substantial rights. Either way, it’s an assignment of the invention.
A written instrument that grants the exclusive right to make, use, and sell an invention acts to assign the invention. It doesn’t matter what the document is labeled–it is what the document does that matters. The document could be labeled “Assignment” or “Exclusive License Agreement” or “Fresh Fish Today.” Doesn’t matter.
Bayh-Dole and its standard nonprofit patent rights clause prohibit nonprofits from assigning subject inventions, other than to an organization that has as a primary function the management of inventions, unless the nonprofit gets federal agency approval. See 35 USC 202(c)(7)(A) and 37 CFR 401.14(k)(1).
But for any such assignment of a subject invention, the assignee is subject to the same nonprofit patent rights clause:
provided that such assignee shall be subject to the same provisions as the contractor
When the assignee becomes subject to that same nonprofit patent rights clause, it becomes a party to the funding agreement. The assignee takes on the obligations of a nonprofit contractor with regard to subject inventions. Work through the definitions of funding agreement and contractor at 35 USC 201.
None of this has anything to do directly with patents.
A nonprofit may assign an invention and for whatever reason insist on retaining title to a patent that has issued or may issue on the invention. That’s not an express concern of Bayh-Dole, which focuses on inventions. Retaining title to a patent has nothing to do with compliance with Bayh-Dole’s standard patent rights clause regarding the assignment of subject inventions.
University administrators make a big show about not assigning their patents on subject inventions, all the while routinely assigning the subject inventions without federal agency approval.