I am looking for an idiom describing a close acquaintanceship or getting on well with someone other than "like a house on fire'.
I've never heard "like a house on fire," so maybe I don't quite know what you mean. That said, a common idiom for quickly coming to a good relationship with someone would be "hit it off," as in "Joe and Martha just met tonight, but they really seem to be hitting it off."
[Edit]: Some others came to mind for me. Two people who are inseparable might be described as "attached at the hip." Also, two close individuals, particularly if they are similar in many ways, may be called "two peas in a pod."
The Cassell Dictionary of Cliches (1996) confirms that "get along like a house on fire" means "to get on together extremely well." A shorter (and less flamboyant) expression of the same idea is "They get along famously [or swimmingly]." Yet another is "They're hand in glove," signifying a close fit or match of interests between two people. Or you could say that they are "birds of a feather."
Another possibility is to say that the two people "take to each other like a duck to water" or that they "go together like a horse and carriage" (or like bread and butter, or peanut butter and jelly, or salt and pepper, or milk and cookies, or macaroni and cheese, or spaghetti and meatballs, or any complementary pair of things, really).
A less complimentary term for closeness is "thick as thieves."
hit it off
get along famously .
From Forrest Gump: "We goes together like peas and carrots."
Meaning: I love you, we belong together, we go well together.
Hot and heavy
If something or someone is hot and heavy, they are full of strong emotions or sexual feelings:
Seinfeld usage: Elaine dates a jazz saxophonist named John Jermaine and admits to Jerry that she has reservations about their dating, because John "actually, he, um, doesn't really like to do 'everything'" (a reference to oral sex). In spite of this, Jerry incautiously tells one of the band members, Clyde (played by Leonard Lightfoot) that Johnny and Elaine are "pretty hot and heavy."
When Jerry tells Elaine what he said, Elaine is alarmed and shows some uncharacteristic considerateness: "I don't want John thinking that I'm hot and heavy if he's not hot and heavy. I'm trying to get a little squirrel to come over to me here. I don't wanna make any big, sudden movements. I'll frighten him away!"
like two peas in a pod
Versions of the phrase "two peas in a pod" date back to as early as the 16th century. In Tudor England, "pease" was the singular form, with the word "pea" not coming into use until the 17th century.
Two peas in a pod is commonly used to describe best friends, twins, siblings, lovers or people in a close relationship who share likes and dislikes.