What is a word for a taking a non-romantic relationship to the "next level"?

Relationships that become stronger are said to deepen.

I think 'becoming/getting closer' [to each other] could describe it sufficiently, without any implications about whether it's platonic or not.

Escalate is often used to characterize the act of increasing the intensity of something.

Bonding comes to mind, usually used in a familial context, but sometimes for friendships as well. It would seem odd to apply it to a romantic relationship, unless that relationship is being planned out ahead of time.

The thought is interesting because everyone recognizes that "friend" covers an immense spectrum yet we have few explicit demarcations. "To grow closer to someone" is commonly used and the closest match to your concept I can think of. Another very common way of expressing this experience is something like "we've been spending a lot of time together"/"I've been hanging out with X more"/"I've been getting to know X better" especially followed by a positive statement about the person or relationship ("and I can talk to him about stuff I actually care about"/"and we get along really well"/"and she's pretty cool").

But it is sort of roundabout. Honestly your description, "becoming better friends", sounds good as is and I'm sorry to say I don't think we have anything fancier.

Referring to time ("old friend"/"family friend"/"childhood friend") can be used to imply the closest kind of friend because a friendship that has withstood time has presumably lasted for a reason, plus you've had the time to get to know each other and strengthen the relationship.

Relationships can be "strengthened", although almost always in past tense, rather than describing an ongoing, gradual change (one rarely hears "our friendship is growing stronger"). It's most often used when overcoming some difficulty raised the relationship up a level. Perhaps your mom was sick and X "was there for you", helping you during that time; or someone spread a mean rumor and X didn't join in the gossip; or you hadn't spoken to X in a while but when you reunited, it didn't feel as if time had distanced you. When this happened "it strengthened our friendship". (To quote J K Rowling: "There are some things you can't share without ending up liking each other, and knocking out a twelve-foot mountain troll is one of them.")