I need help simulating this effect in Adobe Illustrator, but with a different logo, which is also based on stroke. I've tried blending the logo with another external line. Didn't get the same effect.
If using blend doesn't work for you, you can use Offset Path to achieve the same result.
Select the shape and go to Object -> Path -> Offset Path.
You can check the preview option and select the required offset, then click OK.
You can then repeat the process as many times as needed. Offset path will leave the original shape as well as creating the new shape and the new shape will be already selected. The Offset path options will stay the same too, so all you need to do is reselect Offset Path and click OK as many times as you need.
A few notes: For it to work like this, the shape will need to have a stroke (not an outlined stroke). The resulting shape will exaggerate the original shape, so for a shape with rounded corners the result will become more circular. For a shape with hard corners, the result will become more sqaure, like this: