The only thing I was able to find while doing some quick research on VS Code User Guide was that in the Settings editor you can change the tabWidth either to shrink or to stay large enough to view the file name.
// Controls the sizing of editor tabs.
// - fit: Always keep tabs large enough to show the full editor label.
// - shrink: Allow tabs to get smaller when the available space is not enough to show all tabs at once.
"workbench.editor.tabSizing": "fit",