Why must hotel customers check out after a stay longer than a rather low number of days in the United States?

This appears to be related to the COVID eviction moratorium:

[I]t is critical to protect tenants and residents of traditional dwellings from homelessness, as well as those who have lawfully occupied or resided in less traditional dwelling situations for 14 days or more, whether or not documented in a lease, including but not limited to roommates who share a home; long-term care facilities; transient housing in hotels and motels; “Airbnbs”; motor homes; RVs; and camping areas ...

As to why this measure was taken in Seattle but not in California, I can only speculate. Seattle has experienced a higher-than-average level of homelessness over the past several years, in no small part due to soaring real estate prices. It may be that a significant number of people in Seattle rely on cheap hotel/motel stays for temporary shelter when they lose their homes (due to rent increases, eviction, condo conversions, etc.), and it was felt that evicting such people would be bad for public health. But again, this is just speculation on my part.

It appears that in "normal" times in Washington State, the line between "transient accommodations" such as hotels and actual full-on rentals (with the additional protections so implied) is 30 days, not 14. See WACS 246-360-001, which defines transient accommodations as any facility offering stays to guests for less than 30 days.

Because hotels are in the short-term rentals business. If they let you stay too long, they risk turning into a landlord as some states will consider you to be a tenant after only 30 days. Becoming a tenant gives you numerous additional rights so the hotels are restricting the maximum duration of stay. For California see Cal. Civ. Code § 1940.1

Section 1940.1 - Requiring occupant of residential hotel to move or check out and reregister to maintain transient occupancy status

(a) No person may require an occupant of a residential hotel, as defined in Section 50519 of the Health and Safety Code, to move, or to check out and reregister, before the expiration of 30 days occupancy if a purpose is to have that occupant maintain transient occupancy status pursuant to paragraph (1) of subdivision (b) of Section 1940. Evidence that an occupant was required to check out and reregister shall create a rebuttable presumption, which shall affect solely the burden of producing evidence, of the purpose referred to in this subdivision.

From this we learn that:

1. If you stay in the hotel for more than 30 days, you gain tenant rights which makes it difficult to evict you on a short notice. As such it is prudent for hotels in California and other states with a 30-day landlord cutoff to have a limit under 30 days.
2. If you let someone check-in for 30 days, check out and then check-in again, you risk being accused of trying to be a landlord without providing proper rights to your tenants. Hence hotels are requiring you spend a few nights elsewhere before coming back, making it absolutely clear that no landlord-tenant relationship exists between you two.

During the pandemic these laws were made even more restrictive, which is why in Seattle the limit is under 14 days as outlined in the other answer.