A word for "intimate friends" without any kind of romance?

I would say they are dear friends or close friends.

Platonic is the usual choice to describe this.

E.g. "They look intimate but they're merely platonic friends."

Platonic at Merriam-Webster:

pla·ton·ic adjective \plə-ˈtä-nik, plā-\
2 a : relating to or based on platonic love; also : experiencing or professing platonic love
b : of, relating to, or being a relationship marked by the absence of romance or sex

Platonic love at Merriam-Webster:

platonic love noun, often capitalized P
2 : a close relationship between two persons in which sexual desire is nonexistent or has been suppressed or sublimated

First expression that comes to mind is that they are bosom buddies, which the ODO article defines as:

A very close or intimate friend

The relationship between bosom buddies is, despite the fact that ‘bosom’ refers to breasts, quite platonic.

If there is something a bit secretive and reclusive about their friendship, like they’re so close nobody else can really ‘get in’, you might even say that they are thick as thieves:

(Of two or more people) very close or friendly

While bosom buddies is colloquial and informal, thick as thieves is crossing over into very colloquial or even slangy territory, so it might not fit your needs. There is also something a bit disapproving about thick as thieves, which is not the case for bosom buddies.

Other than platonic, I would say "kindred spirits" is an apt term to describe two intimate, but not romantic friends.