I tried looking at this problem by restricting the definition of what a 'rule' is. In my analysis, I consider the case where house order does not matter, and the only rules that are allowed are rules that can be expressed as 2-tuples. For example, rules like (Brit, Red), (Swede, Dogs), (Dane, Tea) or (Green, Coffee) are allowed, while compound rules involving relative positions are not allowed.
For a general puzzle size with $n$ people and $m$ attributes, call the minimum number of clues required to guarantee a unique solution $f(n,m)$. I show that $f(n,m) = nm - g(n,m)$, where $g(n,m)$ is the solution to the following extremal combinatorics problem:
What is the size $g(n,m)$ of the largest multiset of non-empty subsets of $[m] = \{1,\ldots,m\}$, say $\mathcal{F} = \{C_1,\ldots,C_k\}$ such that there exists a unique partition $\mathcal{P} = \{A_1,\ldots,A_n\}$ of $[k]$ with the property that $\sqcup_{j \in A_i} C_j = [m]$ is a partition of $[m]$ for all $i=1,\ldots,n$?
I was able to determine the exact value of $g(n,m)$ for very large $n$, and some relatively tight bounds for smaller $n$. If you're interested, here is the arXiv paper for additional details!