Answer
The web client does support genuine multi-select LOVs, but it isn't a Page Designer toggle — the underlying entity attribute itself has to be configured server-side to support multiselect, at which point the field shows a distinct multiselect indicator and its lookup opens a detail dialog for picking several values at once; a normal single-select lookup attribute can't be converted to multiselect purely through page configuration. Separately, a semicolon-separated "A; B; C" OR-search syntax works in any field's search box regardless of whether the underlying attribute is multiselect — that's a distinct free-text search-syntax feature, not the same thing as a true multiselect field, and it can't be forced onto the picker dropdown itself.