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I thought that paralysis time was consistently the same for units with the same object, but in this game I notice that 2 forest giants froze for different lengths of time.

http://gateofstorms.net/games/57433

Follow Lizard King's forest giant, eventually it gets trapped against the south river by arzenic's forest giant and pus ghol. The pus ghol hits lizard king's forest giant, paralyzing him. Arzenic's forest giant comes up and punches lizard king, but is then paralyzed by a river wight. You would think, based on the series of events, that lizard king would come out of paralysis first, but arzenic does.

Is paralysis really random, and only loosely based on the object tag?

The base paralysis duration is random between 5-12 seconds, then that gets multiplied by the object tag's effect modifier.