"Shadowing" sounds like something that no company would allow, for security reasons. Maybe a startup would not be concerned, but startups are at work 60 hours a week.
Suggest, instead, that you befriend DBAs and chat with them frequently.
There is also "DBE" - Database Engineer, which is sort of a cross between application programmer and DBA.
And "SE" - Service Engineer, which involves keeping the hardware running, and, to a lesser degree, keeping the software running.
If you join a startup, you are likely to wear all those hats, plus make the coffee.
A DBA might do some "programming" -- writing scripts to handle backups, making schema changes, applying changes that the programmers ask for, etc.
A programmer is more involved in the Business logic of the application; a DBA is more involved in the details of the database, and may be somewhat ignorant of the application.
When (on this forum), I say "do this" (as a DBA), I want to know enough about the application to avoid advocating a cookbook reply that does not apply in their situation. So, I think of myself as a DBE.
Read the threads in the "Best of the Best" in
http://mysql.rjweb.org/bestof.html
They give you some insight into what a DBA/DBE needs to know and say. (Many, not all, were authored by me.)