The statute is the result of Senate Bill 882, which was passed by the legislature and signed into law in 1999. The whole bill can be read here.
You'll notice that the bulk of the bill does not deal with transitional housing campgrounds but rather with easing state zoning regulations in rural areas in order to allow private recreational campgrounds to build yurts. Why would that be? Here's my theory:
Note that the bill is sponsored by the Committee on Agriculture and Natural Resources "at the request of Sen. Veral Tarno." Based on a very cursory Lexis search, it seems that Sen. Tarno, a Republican representing a rural district, was an opponent of Oregon's land use and planning statutes, particularly the state's landmark 1973 legislation that aimed to prevent sprawl and preserve agricultural lands. For example, in an AP article from 1/31/99, Tarno is quoted supporting a bill that would ease residential building restrictions on agricultural land in Coos County, asserting that land use regulations at the time were stifling economic development. It also appears that the 1999 legislative session saw a slew of bills intended to weaken the 1973 statutes. The director of 1000 Friends of Oregon, a land use watchdog group, is quoted in the same article claiming that at least 70 bills had already been introduced that would weaken statewide zoning and allow more unrestricted building in rural areas.
So why yurts, why campgrounds, and why Sen. Tarno? Apparently, under the 1973 planning regulations, yurts could not be placed on private campgrounds in some rural areas. And this is precisely what a constituent of Tarno's wanted to do.
In the minutes of a 5/19/99 meeting of the House Committee on Water and Environment, Sen. Tarno speaks in support of SB 882, saying it is necessary for the completion of the "Canopy Project" (minutes are here). Also see discussion of the bill in the Senate Committee on Water and Land Use here. The "Canopy Project" seems to have been a tourist development project in Curry County, controversial because it involved not only development of hitherto protected wilderness of ecological importance (e.g., in a park with redwood stands) but development of public lands by private corporations. See this letter from various environmental groups to the state's Parks and Recreation director in opposition to the plan:
Now onto the issue of transitional campgrounds and yurts within urban growth boundaries...
This seems to have begun independently of Sen. Tarno's SB 882, in a bill (HB 3082) introduced by House Rep. Al King, a Democrat (who I believe was representing a district that included Eugene). You can see the bill's history here. Note that future Senator Jeff Merkley was a co-sponsor. After HB 3082's initial reading on the floor, it is referred to the House Committee on Agency & Performance Operations. At its first committee hearing on 3/30/99, King introduces the bill and describes its purpose as providing a low-cost option for people who would otherwise be homeless. During public comment, the director of Oregon Housing and Community Services speaks in support of the idea. But much more interestingly, another speaker in support of the bill is a man named Alan Bair. A constituent of King's, Bair is the owner of Pacific Yurts, Inc. (minutes). Bair also offers testimony in support of HB 3082 at the committee's meeting on 4/6/99 here.
At this point, Rep. King's HB 3082 seems to have dissolved into an amendment to Sen. Tarno's SB 882, which is by now in the House Committee on Water and Environment. The 5/19/99 meeting of the committee is where the amended SB 882 is introduced (minutes). The strange amalgamation of the two bills would explain why the bill that ultimately passed seems to have two different definitions of "yurt": one for the parties who want to develop for-profit eco-tourism in state parks, another for parties interested in creating transitional campgrounds in urban growth zones. During the committee's 6/2/99 meeting, Rep. King refers to these separate definitions as a "firewall" that separates the first sections of the bill (which have to do with Sen. Tarno's original bill) from section 6 of the bill (which would ultimately be incorporated into statute law as ORS 446.265).
The Department of Land Conservation and Development did not support Sen. Tarno's original bill. But at the committee's 6/4/99 meeting, the Department reports that their concerns have been addressed in earlier amendments (section 3 of the final bill). By now, the bill is ready to leave committee and head out for a final vote. Jeff Merkley himself motions to adopt King's amendments to SB 882. The motion passes and the committee sends the bill to the floor with a "Do Pass" recommendation. After moving to adopt the amendments, the minutes report that Merkley "notes an experience he had with a homeless family. Comments on the benefits of yurts as opposed to shelters." Perhaps Sen. Merkley is an untapped supporter for Opportunity Village and similar organizations.... (minutes).
So to answer your question whether other states have similar statutes, my guess is no. ORS 446.265 is the product of a Frankenstein bill with the unlikeliest coalition of supporters (eco-tourism developers, social services administrators, homeless advocates, and yurt manufacturers) and opponents (environmental groups, planners, and rural residents concerned with population growth).