In history, their alliance was practically unbreakable and in the show, it's represented as a matching unbreakable friendship.
The "Bad Touch" comes from sadly, their unmistakable somewhat perverted and childish behavior.
In the series, France is stereotyped as perverted and overly romantic because he's the "country of love". You'll most likely find the man flirting with people from random pretty girls to countrys like himself which consists of England, mostly because he likes to annoy the crap out of him.
Spain is stereotyped as a warm, always happy, loving parent but you should never let your guard down around the Spaniard. If given the chance, you could plainly see that he's the series runner up for "Most Romantic"
Last, but not the least, Prussia is stereotyped as the extremely cocky, self proclaimed "King of Awesome". When he was a ruthless country, back when he was one, and always referred to his growth in power as "Being Awesome". The suggestive part about him is that he exclaimed his conquests of land as "Seizing Vital Regions!" Pretty much enough said. (He lives in Germany's basement now so don't get your hopes up.)
A fan-notion added to the theory of this friendship is that the trio found each other's laughs absolutely hilarious.
England: It must have been the Bad Touch Trio again ... bloody hell! They just stole my favorite union jack boxers!
France: Oh hon hon hon~
Spain: Fu so so so so~
Prussia: Ke se se se se~