I recall hearing a rationale explaining that Kacie Jane has it right, but the funny thing is that where I had this explained to me was in Montreal. In French-speaking areas the "street type" always comes first: boulevard Rene-Levesque, chemin du Village, rue Sherbrooke, avenue du President-Kennedy. The reason it's that way is that in French the common noun normally comes first and the adjective comes second—for example, we say "red wine" but they say "le vin rouge" (there are exceptions to this general principle). Thus, the one particular street is designated a "road" and then the adjective coming afterward tells you which
road (Sherbrooke), with the "generic" or "common noun" status underscored by the fact that they don't normally capitalize the type of road if another word comes before it.
Of course, the rules of French grammar don't necessarily automatically follow into English, but I find the explanation to be quite logical. Obviously it's not unprecedented for English-speaking areas to put the street type first—consider all the lettered avenues in Brooklyn, for example (such as Avenue X).
The one that threw me for a loop when I heard it is Avenue Road in Toronto—I wondered whether it is an avenue named Road or a road named Avenue (apparently the latter is correct).
Anyway, returning to the topic:
I've seen the independent cities in Virginia use these sorts of signs both on entry and exit, although often they're used as adjuncts to the "official" boring green VDOT-issued "Enter/Leave" signs.
Oddly, here in Fairfax County at least one of the county's administrative districts posts some fancy signs of that sort when you enter and leave the district. The Bing Maps "Streetside" view is a lot clearer than the Google Street View image.
I pass this one almost every day. It has the Fairfax County seal on both sides and it tells you you're entering or leaving Lee District (the side shown here says "Drive Safely" and the other side says "Lee District Welcomes You"). I've never seen that level of specificity anywhere else!