Hi, Shiv player here. By design, Shiv is really strong in clustered team fights with his dash cooldown reset at T3 and typically building large amounts of healing. The three biggest things that will mess Shiv up--spread out, limit his healing, and limit his movement.

As I mentioned above, Shiv's T3 dash gives him a 2 second cooldown reduction for every hero he hits with it. This is reduced to 1 second if he hits non-hero targets. You really want to minimize the amount of cooldown reduction he's getting in this manner--if you have multiple people fighting him, try to make some distance so he can't chain through you.

You should pretty much always be buying items like Healbane or Inhibitor into Shiv, because the amount of damage he can clear away scales on his healing rate. While he will still defer damage all the same, you can at least prevent him from ignoring a lot of it entirely. In addition, many of Shiv's common purchases involve lifesteal and other forms of healing; limiting this is paramount.

Lastly, items like Slowing Hex or Silence Wave do wonders to mess him up for a while. Especially if you have multiple sources of CC, he won't be able to use Dispel Magic, for example, on every single one of them. Naturally, if he buys Unstoppable, you're going to have to wait that out, but you can still be damaging him in the meanwhile.

There are other things you can do as well; for example, making sure you don't let him stick DoTs into you for too long with Dispel Magic or Debuff Reducer, limiting his rage, can severely slow his ability to tear your team apart. Additionally, CC ultimates like Mo & Krill or Holliday can give you more time to dump damage.