This can be done with straight connectors or with right angle connectors. It's the gluing method/technique that is important.
The macro below is an example of gluing right angle connector between two shapes. It should be noted here, that, a straight connector will literally go between the glue points on each shape. So, if they're identical, the connectors will overlap.
Fundamentally there are two gluing methods. 1) letting Visio decide where best to put the "apparent" glue point. This is typically called walking glue. 2) specific glue points are provided. These points can be determined by either using connection points, or by specific geometry points (with glue to shape enabled). The provided code / example takes the beginning end of the right angle connector and glues it to a point on the left side of shape1. This is method 2. It then takes the "ending" end and dragged it over the center of shape2 until the entire shape highlight in red. This indicates walking glue and is method 1. The resulting route looks like a straight line. If this is repeated for a 2nd connector, except with a different start point on shape1, you get a 2nd, straight appearing, non-overlapping connector. However, if you then convert the two connectors to straight connectors, the left, beginning ends stay put, the right,endings move, point toward the center of shape2. The end points move because the literal connection is center of shape2.
In the attached example, the upper pair used the outlined process. The code pertains to the blue connector. If you open the shapesheet, you can see the syntax for beginning and ending points is different, indicating the different glue methods.
The lower pair is just a copy of the upper pair, but the connectors were changed to straight.
The code for a single connector, cleaned-up from macro recorder:
Sub Macro2()
Dim vsoCell1 As Visio.Cell
Dim vsoCell2 As Visio.Cell
'Glue beginning to specific location on shape, not using connection points
Set vsoCell1 = ActiveWindow.Page.Shapes.ItemFromID(3).CellsU("BeginX")
vsoCell1.GlueToPos ActiveWindow.Page.Shapes.ItemFromID(1), 1#, 0.75
Dim vsoCell3 As Visio.Cell
Dim vsoCell4 As Visio.Cell
'Gluing connector end to center of 2nd shape:
Set vsoCell3 = ActiveWindow.Page.Shapes.ItemFromID(3).CellsU("EndX")
Set vsoCell4 = ActiveWindow.Page.Shapes.ItemFromID(2).CellsSRC(1, 1, 0)
vsoCell3.GlueTo vsoCell4
End Sub
Note, if you drag the upper left shape about, say upwards and downwards, its connection points stay the same. But, the connections to the right shape will move, i.e., walk around the outer edge of shape2. Plus, as they walk, the red and blue connectors remain separate, spaced by the separation established on shape1
Wapperdude