Only getting 1 dimension

Blur2's Viewport should also contain a TextureRect. It should display the ViewportTexture that is produced by Blur1's Viewport.

So the order of operations would be something like:

1. TextureRect is drawn inside of Blur1. The TextureRect is the one that has the first blur shader, so it's drawn with the first blur pass directly into the viewport. It has whatever texture (in your case the Godot icon.) Its ViewportContainer is invisible (self-modulate's alpha set to 0)
2. TextureRect is drawn inside of Blur2. This texture rect uses the ViewportTexture created by Blur1. Either it, or its ViewportContainer, can have the second pass blur shader. Its ViewportContainer is visible, so it displays the result.

The alternative is to have nested viewports (Blur1's ViewportContainer is a child of Blur2, and you use the ViewportContainers to blur.)