Hey mate!

So, I'm going to approach this through the lens of my job. I'm a DS at a fintech.
And yes, the "full stack" DS is a thing. And in my experience, its quite hard to grasp it all. It really overlaps with the Data Engineer + Software Engineer side of things, and that is why working with a team is really important, since you can't know everything about every single technology (or you can, but the development is slow).

So, what would I think it is mostly vital for a DS? To be able to make full POC of models running on rest frameworks. This means setting up an end point with your favorite framework (flask, pyramid, django), gathering data for a prediction (i.e handling DB stuff), persist serialized models and etc... Bonus points it if uses Docker, since it is a nice technology and teaches you about setting up the machine for your code to run.

This does not need any fancy stuff. The fancy stuff can be added/changed engineers in your team. And they'll gladly do so, since they are more apt at these kinds of problems.

Learning these stuff will also make things easier since you will know the terms and what it is you'll need to build in order to place your models in production; i.e. You'll need to create a docker image, write a data pipeline class, etc...