There appears to be an incompatibility between this theme and sphinx tabs.

conda create --name example
conda activate example
conda install pip 
conda install sphinx pydata-sphinx-theme --channel conda-forge
mkdir test
cd test
sphinx-quickstart

Setup the conf.py to use this theme and the switcher and then make the docs

make html

Then navigate to the docs and you should see the dropdown.

Now, if you install sphinx-tabs and load the extension (even without using the tabs) the switcher disappears.

conda install sphinx-tabs

Add the extension to conf.py

extensions = [
  'sphinx_tabs.tabs',
]

and remake the docs

make html

Refresh the docs page and you'll see that the switcher doesn't work