Bloom has an Auto-save feature that allows you to jump back into the visualization you were previously working on.
You get the option to restore the previous visualization when you log back into Bloom from the same browser.

It doesn't have the option to save or restore scenes manually.
Also, since the visualization is stored in your web browser, you won't get the restore option if you use a different computer/browser or if you clear your cache for Bloom on the same web browser.

Limitations with Neo4j Aura

Neo4j Bloom Enterprise allows Saving and sharing scenes. But Bloom Basic doesn't.
Aura comes with Bloom Basic and hence doesn't have the option to save and share scenes.

Work Around:

You could utilize the Saved Cypher feature for reuse functionality.
Note:
While the queries saved using the Saved Cypher feature are also stored in your web browser, you can easily copy the cypher text and save it elsewhere for reuse later.