What kind of word is "Frankenstein" in this context?

It's called a denominal verb, according to an article on JSTOR's blog, which discusses examples such as "My sister Houdini’d her way out of the locked closet." The process of using a noun as a verb is called verbing a noun, or (for lovers of long words) denominalization, and is very common in English going back at least to Shakespeare (although using a proper noun is less common than using other types of noun). It seems quite rare for people to use a specific term for the verb, other than using an adjective like "denominal" or "denominalized". Here's another article on the subject, from Grammarly.

The first use of the word as a verb that I have found after a cursory search of Google Books seems to mean "to scare" judging by the context. This dates from 1829, about 11 years after Mary Shelley first published the famous novel.

We've been Freischutted , Frankensteined , They choose a dress of flaming tint , And buy the stuff ell - wide , sirs . And Vampired ' nough t'alarm any ; And if we chance to lay one fiend , Ri doodle , & c . The Universal Songster: Or, Museum of Mirth

Not very long after, in 1842 we see it in its meaning of "to cobble together"

Some padded figure for a pantomime , or an Egyptian mummy , Frankensteined into a kind of counterfeit vitality ! - Suddenly swollen individual ! how many coats have you on ? " " Only four , sir , " replied Mrs . Drabber

The Court Magazine and Monthly Critic

Stuart F has named it a denominal verb, which seems as good as any. Needless to say, the main qualification of a name used as a verb is that the name should be very well known in the community where it is used.