Simple Mask Allows Part of the Bottom Image to Show Through

"If you want, you can call it a display issue or anything else you want to call it, but the fact remains when it exports the image, the fringe red line is visible."

That seems logical. A vector object has to be rasterised to display it on the pixels of a monitor screen and to export it as an image. If there is an issue with the rasterization, it may show also on export.

In your case an opacity mask does not have that problem.

Definitively it's just a display effect or issue. Try to zoom in or out and the red line keeps its width. Or export to PDF and experiment something similar.

As Ton Frederiks points as precisely, it is an anti-aliasing rendering problem, on the display

Start again. Build the peices and then use the top object (ellipse) to create the clipping mask. Everything in that layer will clip to the mask. If you do not want it part of the masked items, put it into another layer. The process is rather simple. The red object is there since you put it there. I assumed you did not want the red object but the logo is clearly red so red has to be there hence I put it on the mask. Sometimes starting over and rebuilding clean is the best way to fix the problem. (I have done the same many times in my life)

ANSWER: What I thought was the answer, rebuiding Illustrator preferences, was not.

Doing more research again.

If you want to keep everything editable, you could try an Opacity mask.

Group everything below your circle.

Select the group and the white circle and click Make Mask in the Transparency panel.

Did you try Pathfinder Merge?

Ok. How about if you do not want the red object on the bottom. Make the Clipping mask red. In this example I used the Draw Inside options. The Clipping mask can have properties like a fill and a stroke. The stroke always sits on top of all other properties and the fill is behind all other shapes. 

Draw Inside Option:

This makes a clipping mask group (not a clipping mask layer)

1. Select the ellipse. (Mine has a color fill and a color stroke)

2. Select the Draw Inside button in the tool panel

3. Deselect the selection.

4. Change the properties of the next shape to be drawn and draw the shape. The new shape will clip to the Draw inside shape. 

Changing properties of a Clipping Mask after it is created:

If you use the Make/Release Clipping mask option   in the layers panel or use Object>Clipping Mask> Make, you will make a clipping mask but without properties. You can always choose the clipping mask and add properties back in after.

1. Select just the Clipping mask: Target or select it with the Group Selection tool.

2. Change the properties. (Fill and Stroke)

The fill can be covered up by other fills and strokes inside the clipping path but the stroke on the clipping path is above all other properties.

Does this sound like more what you are trying to do?

The image shown was created was two rectangles of two different colors (red and white), with a elliptical clipping path to create the final object. 

If you are creating a logo, then you probably do not need a clipping mask. Use the Pathfinder panel instead. You can use just the ellipse and one rectangle to accomplish the same goal. If you keep the rectangle on top then select both the ellipse and rectangle, you can use the Minus Front button from the Pathfinder panel which in this case would create a new simple path. If you wanted more freedom, hold the option/alt key when you choose the button. This will create a clipping mask which let's you change the object more easily after the fact since the two original shapes are there in the object.

I see what you mean.

Looks like an anti-aliasing rendering problem.

Pathfinder Merge or Export as an image with Art Optimized seems to get rid of it.