Are there computer graphics programs that allow use of fractional units (e.g., 1⅜") instead of decimal (1.375")?
The BIM / 3D / draughting software ArchiCAD supports fractional unit entry in a similar manner to that described with SketchUp, recognising a typed entry such as 9-2 1/2 as 9'-2 1/2" - the dash is the divider between feet and inches, the space indicates the gap between an integer and the fractional portion of a given dimension: of course, ArchiCAD only does this when configured to US/Imperial units for a given project - in metric it prefers decimal input.
FWIW - Both Adobe Illustrator and Affinity Designer will recognise single element fractional entries - it parses these as a maths string, so if you enter for example 2/5 into the Width unit in the Transform palette, (and your Document Setup has identified Inches as your unit) it will transform that dimension in your geometry to the correct 0.4" size - but it cannot for example parse 1 1/2 - hence my single element constraint. The maths parser means that you can, for example, enter 0.4*3 and the transform will give you a length of 1.2".
The program known as SketchUp will accept fractional dimensions when creating models. Depending on your application, this program may not be suited to your purposes, as it is more frequently used for architectural drawings. Perhaps adding an indication of your use would be helpful, but SketchUp is one answer, as is AutoCAD and similar CAD programs.
In such applications, the information is entered as 1-5/8 for your example. I've confirmed this with both SketchUp and AutoCAD by a quick 'net search and have used this method in the distant past.