Yes. You can save a custom sort for the content within a board by selecting "Save View" after creating it. This saves the sort as that board's default view for people who visit the board, including through its share link or public URL.
Saved board sort does not control the order of boards or sub-boards in the left sidebar, and it does not control what appears in Favorites. Sidebar ordering and Favorites are managed separately from a board's saved content view.
This also does not create a workspace-wide default for all new boards, and Search all files does not use the same saved-sort behavior. For sidebar navigation controls, see Explore Workspace Sidebar.