Charcoal, 27 23 bytes

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υＩＥ⊕⌈υ⌕Ａυι

Try it online! Link is to verbose version of code. Explanation:

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υ

For each element in the final diagram, convert it to base 4, halve the digits, then convert from base 2. This gives the row number for that element.

ＩＥ⊕⌈υ⌕Ａυι

For each row list the elements assigned to that row.

Charcoal, 27 23 18 bytes

≔ＥＸ²Ｎ↨⁴↨ι²υＩＥ⊗υ⁺ιυ

Try it online! Link is to verbose version of code. Explanation: Now a port of @tsh's Python answer.

≔ＥＸ²Ｎ↨⁴↨ι²υ

Map the numbers up to 2ⁿ by converting them to base 2 and then interpreting the result as base 4.

ＩＥ⊗υ⁺ιυ

Vectorised add the doubled array to itself as a matrix.

Previous 27 23 byte answer uses a different approach that is interesting in its own right because some golfing languages have built-ins for things like group index by value or sort array by key.

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υＩＥ⊕⌈υ⌕Ａυι

Try it online! Link is to verbose version of code. Explanation:

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υ

For each element in the final diagram, convert it to base 4, halve the digits, then convert from base 2. This gives the row number for that element.

ＩＥ⊕⌈υ⌕Ａυι

For each row list the elements assigned to that row.

Charcoal, 27 bytes

⊞υ⁰ＦＮ≔⁺⁺υυ⁺⁺υυＸ²ιυＩＥ⊕⌈υ⌕Ａυι

Try it online! Link is to verbose version of code. Explanation:

⊞υ⁰

Start with a 1×1 array of row numbers.

ＦＮ

Repeat the given number of times.

≔⁺⁺υυ⁺⁺υυＸ²ιυ

Each time, the array becomes four times larger. The previously calculated 4ⁱ elements go in the top left quadrant so they remain in the same row, the next 4ⁱ elements go in the top right quadrant so they have the same row as the previously calculated elements, then the remaining elements in the bottom left and bottom right quadrants go in the next 2ⁱ rows.

ＩＥ⊕⌈υ⌕Ａυι

For each row list the elements assigned to that row.

Charcoal, 27 23 bytes

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υＩＥ⊕⌈υ⌕Ａυι

Try it online! Link is to verbose version of code. Explanation:

≔ＥＸ⁴Ｎ↨²÷↨ι⁴¦²υ

For each element in the final diagram, convert it to base 4, halve the digits, then convert from base 2. This gives the row number for that element.

ＩＥ⊕⌈υ⌕Ａυι

For each row list the elements assigned to that row.