Set the Instant Pot to Sauté mode. Heat the olive oil, then add the onion, celery and carrots. Sauté the vegetables, stirring occasionally, until starting to soften, 3 to 4 minutes.
Add the garlic, crimini and shiitake mushrooms, thyme and pepper. Cook until the mushrooms are starting to release their liquid, 2 to 3 minutes. Stir in the broth and salt.
Put the lid on the Instant Pot, close the steam vent and set to HIGH pressure using the manual setting. Decrease the time to 10 minutes. It will take the Instant Pot about 10 minutes to reach pressure.
Once the time is expired, carefully release the steam using the quick release valve.
Transfer half of the soup to the blender, add the coconut milk, hold on the top and blend until almost smooth (leave a little bit of texture), stopping the blender and opening the lid occasionally to release the steam. Transfer the pureed soup to a pot or bowl. Repeat with the second half of the soup.
If the soup needs to be reheated, return it to the Instant Pot and heat gently over the Sauté mode. Serve.