No, quotients of polynomial rings are definitely not "almost UFDs".
Any finitely generated ring over $K$ is such a quotient and this means a lot of non UFDs. Said differently, any algebraic variety in affine space over $K$ has as ring of regular functions one of your quotients and in general (as your own example over $\mathbb R$ states) it will not be a UFD.
By the way, your parenthetical remark about the complex case is a bit ambiguous: $\mathbb C[x,y]/(x^2+y^2-1)$ IS a UFD: by the change of variables $u=x+iy,\ v=x-iy$ this ring becomes $\mathbb C[u,v]/(uv-1)=\mathbb C[u,1/u]$, which is factorial and even a PID. (Reason: If $A$ is a PID, so is every ring of fractions $A_S$ [...unless it is a field, you Bourbakistas].)
