This isn't a full answer, since it depends on the laws and regulations of the place of employment, including university contract rules. However, note the following.
There is a big difference between holding a patent and monetizing it. The latter is a business process, not a research process, and takes lots of inputs. Money doesn't automatically flow from the possession of a patent.
Obtaining a patent in the first place can be expensive since there are (varying) regulations governing it that normally require lawyers (plural perhaps, expensive usually).
Defending a patent against someone who wants to invalidate it can be very expensive. Even defending against someone who wants to exploit it without permission or license can be very expensive.
For university employees, as for employees elsewhere, there are often contract stipulations about patents. A typical contract provision gives the responsibility for obtaining, licensing, and defending a patent to the university itself. This is normally advantageous due to the costs entailed. Large research universities may have patent attorneys on staff or on retainer for this.
The patent may be applied for in the name of the researcher, or (perhaps), jointly in the name of the researcher and the university. Other contract arrangements are possible, and I don't know all of them. The researcher may be entitled to a portion of any profits that derive from the patent, but are normally protected against losses. The last is, potentially, a big deal.
I don't know of funding agencies who get involved in this, but that isn't outside the realm of possibility.
For your own case, if you are in a situation where it matters, start with your own employment contract. Or, consult a patent attorney.
While the patent may remain in your name only, you may be required by contract to assign some of the rights to the university (or company). Search for "Patent assignment" on google or other search engine. An example explanation is here: https://www.legalzoom.com/articles/patent-assignment-how-to-guide