IIRC Threads is 500 characters, not 240, so that wouldn't be a something they could sue over.
However, Twitter has over 2000 patents filed, over 1400 of which are registered in the United States. They deal with malicious advertisement detection & remediation, malware, automatic management of data processing, mobile app management, interactive advertising, and electric publishing to name a few.
I can see where Musk may have found similarities between the software somewhere in the mess of their patents.