Why was Fort Sumter important to abraham lincoln and Jefferson Davis?

It was the big challenge.

Was this tiny island-fort part of the US Army, or was it part of the newly-formed Confederacy?

Would the Confederacy dare to attack the might of the Union?

And if so, would Congress dare to make an aggressive response (equivalent to the war it couldn't declare because it didn't recognize the Confederacy)... by appealing for new volunteer troops?

When the fort surrendered, Lincoln called for recruits to the Union Army, because it appeared that a fighting war was unavoidable.