This isn’t really about the word share, per se—the word is being used in a way here that is specific to a certain field, essentially jargon—but more about perspective.
A share is normally just a part of something, seen at a relative size; imagine a share of a cake. In this normal usage, shares can be bigger/larger or smaller. Such a share may be said to represent a certain percentage of something (a 50% share is bigger than a 30% share).
When talking about housing expenditure share, however, the word takes on a slightly different and more absolute meaning, which is semantically closer to rate or the likes. Such a share is also measured in percentages, but it is not seen so much as a relative percentage of a whole, a chunk cut out of the whole ‘cake’. Rather, this type of share—like rates, levels, and other types of values—is conceptually imagined as a numeric value on a scale between 0% and 100%, so just like your blood oxygen levels can be higher or lower than someone else’s (despite being measured as, theoretically, the relative share or portion of oxygen saturation in the blood), your country’s housing expenditure share can also be higher or lower than another country’s.
So whether you use bigger/smaller or higher/lower depends on how you conceptualise the value that you’re talking about.
Note: I am not an economist, and I don’t understand the finer points of such economy jargon, but it should be noted that housing expenditure rate also yields hits on Google (about 25 vs. about 95 for housing expenditure share).