No - there is no chance of any school finding out at all about other schools you have applied to, even using the same system. Take another example. You could use Paypal to pay many merchants (using the same portal), but one merchant will never know you are using the portal to pay another merchant too. Similar is the case with B-schools. They may use the same application system, but have no visibility into the user's activity on the system not pertaining to their own use of data.
Thank you so much for replying. That's good news