How about two people pulling on a spring? It seems to me that you can’t push or pull on someone without feeling the same push on yourself. The person who remains standing is the one who can better oppose that force. But both feel the same force.
Sure they can, because the rope can accelerate in either direction (thats how you win the game!).
Imagine two weights $m_1$ and $m_2$ on rope s.t. $m_1>m_2$, both masses feel the same force of tension, however the rope accelerates towards $m_1$, and in turn the whole system accelerates. The same is true for the tug of war.