'share a common thread' is correct and can be used in written English.
You can use it to describe a situation in which two or more different things have something in common. For example: "The novels of this author all share a common thread; they all explore the nature of freedom."
Their answers didn't share much commonality in terms of industry, but they did share a common thread: The expanded or strengthened use of existing algorithms.
Many of the lesser-known acts share a common thread of amalgamation.
His recent defeats share a common thread: blown leads and missed opportunities.
These stories share a common thread: due diligence of environmental impacts attributed to debt financing.
The three states share a common thread: President Obama pulled them into the Democratic column in 2008.
They do share a common thread as two of six N.F.L. teams that have chosen not to have professional cheerleaders.
He said the group's raps "shared a common thread with protest music".
Mr. Yoshikawa, who grew up in an old Japanese temple house, said his father had pointed out something that architectural experts also note: "He said Luigi's design, rooted in good Western design concepts, shares a common thread with traditional Japanese homes".
My mother's story of her second battle with cancer, although woven differently in its details, shares a common thread for many.
Although he'd learned from past actions to thoroughly document everything he did, Europe for President shared a common thread with the Guantánamo debacle in its lack of press.
Text comments expressing this belief were generally concise, such as: 'Respect....key principle!' 'Respect....key principle!' Ideas were slightly more developed by other respondents but shared a common thread: 'The patient's wishes with respect to spiritual care should be addressed.
