The OP asked for a more specific word than "arpeggio" to describe a figure.
My answer was converted to a comment, either because it was short, or because I didn't come up with a more specific term.
My answer was "Arpeggios. Covering a full octave, if you want to say more. They are repeated; and they are slurred."
No one has come up with a more specific term yet.
The question in the title was, "What is it called when a violinist quickly bows all strings in an 'arpeggio-like' flourish?" My answer is repeated, slurred arpeggios covering a full octave.
I suppose one could also say that the arpeggio covers all four strings. Concisely: a four-string arpeggio. I have never heard that phrase -- I just made it up -- but it would help the other person to envision the arpeggio.
The last sentence of the post was, "What is the term for this — is there a term for this?"
To the question "Is there a term for this [other than 'arpeggio']?" my answer is No, there is not.
That is my answer. How may I post this without it getting converted to a comment?
My source of information is a Bachelor's in Performance (Cello) from a prestigious music school.