fixes #71671

src/vs/workbench/contrib/debug/common/debugModel.ts

@@ -797,7 +797,7 @@ export class DebugModel implements IDebugModel {
 				// Make sure to de-dupe if a session is re-intialized. In case of EH debugging we are adding a session again after an attach.
 				return false;
 			}
-			if (s.state === State.Inactive && s.getLabel() === session.getLabel()) {
+			if (s.state === State.Inactive && s.configuration.name === session.configuration.name) {
 				// Make sure to remove all inactive sessions that are using the same configuration as the new session
 				return false;
 			}

src/vs/workbench/contrib/debug/electron-browser/debugSession.ts

@@ -783,6 +783,7 @@ export class DebugSession implements IDebugSession {
 		}));
 
 		this.rawListeners.push(this.raw.onDidExitAdapter(event => {
+			this.initialized = true;
 			this._onDidEndAdapter.fire(event);
 		}));
 	}