Oh, OK, as she isn't a long-term friend in real life I would be worried I'm afraid.
XH2 got the hump about HIS best friend (married) and me texting each other with all sorts of inane crap on the evening commute. It so happened that we both had a long journey at around the same time, and were both very chatty types. To be fair, XH2 got the hump about almost everything, and wasn't remotely chatty. He had a good look at the messages and ran his worries past the best friend's wife, after which he was okay with it (if uncomprehending!)
That was different from what you're describing, though. It was a natural extension of a real-life, existing friendship. Your H's relationship seems to be entirely constructed on these texts ... which is odd 