Same issue here! I was unable to download from Blob Storage (storage v2) with latest versions of the Storage Explorer.
I tried versions 1.15.1 to 1.11.0 and ended using 1.10.1 that was finally able to download files.
And although I didn't try it out here's what I'm guessing:
Enabling the AzCopy feature in 1.10.1 could be a flag that is persisted in user settings. So my bet is that when I would now install the current version it would work as well.
So to the developers, @jinglouMSFT, @Sumarigo-MSFT : Please check, if the "Enable AzCopy Feature Flag" is not set to true in versions younger than 1.10.1 and if so, please enable it during installation.
So my solution for now is to use the old version (1.10.1) that comes with AzCopy as an optional feature where I can turn it on and it's working like a charm.