The "rent" in "rent-seeking" does not refer to "rent" it refers to "economic rent."

Totally different concept. But don't take my word for it:

> "Rent-seeking" is an attempt to obtain economic rent (i.e., the portion of income paid to a factor of production in excess of what is needed to keep it employed in its current use) by manipulating the social or political environment in which economic activities occur, rather than by creating new wealth.[0]

> In economics, economic rent is any payment to the owner of a factor of production in excess of the costs needed to bring that factor into production. [1]

[0] https://en.wikipedia.org/wiki/Rent-seeking

[1] https://en.wikipedia.org/wiki/Economic_rent