Shapefiles utilize IEEE 64-bit floating-point representation to store coordinates. They have a precision of at least micrometres at the Equator and 180th meridian using decimal degrees, far in excess of the capabilities of geodata collection. Shapefiles can map the logic gates of modern semiconductor CPUs within a UTM zone, if x-ray crystallography or a scanning electron microscope were used to collect the locations.
When working with global international boundaries, the US State Department "Large Scale International Boundaries" (LSIB) dataset is designed for accuracy (albeit to US policy), and its metadata includes (emphasis mine):
The LSIB is in WGS84 datum and is generally accurate to within a couple hundred meters or better.
If you compare this with your five decimal place coordinate values (precise to ~1 meter at the Equator), you can begin to see that the issue is not in the shapefile format, but the coordinate generalization of the dataset in use.
Large-scale boundaries are slower to render due their storage requirements (often one hundred times larger than their small-scale brethern). Ultra-large scale boundaries would be very difficult to use at most scales, and fantastically expensive to collect (even in places where the boundary is undisputed).
The problem here is not a matter of the data format, but of the applicability of the dataset to the intended use.