They would appear if your analysis is not complete. You can turn off the banner in the UI for the effect.
It is also almost always a good practice to use only the footage you are processing with warp stabilize in a comp, only run warp stabilize, then render a digital intermediate using a suitable visually lossless production codec so you can use that footage in another composition.
If you are compositing something in footage that is warp stabilized, like replacing something in the shot, then it's a good idea to NOT run warp stabilizer on the footage, but instead nest the composite with the tracked composite in a new comp and run Warp Stabilizer on the nested comp instead of the footage. Warp stabilizer is a CPU and system hog, the results bloat your project size, if you run out of cache there can be problems and it's generally finicky. If a shot is longer than 2 or 3 seconds I always create a DI and replace it in the production. I seldom if ever save a project that has a warp stabilized layer included in the project because it's just not reliable.
I will approach it with more caution.
I just found it highly odd that a UI banner for AE appears in a final rendered MP4.
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It is unusual but if Warp Stabilizer has not finished and AE is trying to process the stabilization in the background, which is the way the Adobe Media Encoder works, it makes sense to me that it could appear in a rendered project if the cached frames are lost or never rendered at full resolution. You are correct in calling it a bug. There should be a switch in the plug-in that won't let a render start until the processing is complete. If you got the processing banner in the render then the stabilizing didn't happen either and there's no provision in the software preventing a render from starting before all of the frames are rendered.