If you resided at the (otherwise weekly) motel on a continuous basis for the entire year, we can think of no particular tax reason why this could not, or would not, be considered functionally equivalent to renting an apartment for the year . . . at least with respect to claiming, say, a state-level renter's tax credit / deduction, or some similar tax benefit that may be available to renters.
Legal rights to occupancy and privacy may differ, but neither extended-stay motels nor apartments give their tenants rights to building or land ownership, and the concurrent responsibility to pay for property taxes. Rent is rent; ownership is ownership.
In other words, there should be no (tax code) discrimination applied to a week-to-week motel, when compared to an apartment. They should be treated equally for income tax purposes; and so, in TurboTax, you would indicate that you paid rent, yes, in response to any such software question(s).
Thank you for asking this.
