Choose a different name after converting illustratorlayers to vector in aftereffects

There is no auto renaming of layers. You might be able to write a script that added a suffix to layer names, but I don't think it's worth it most of the time. 

The trick to preparing AI layers for video is in the naming and preparation. Converting a vector layer to a shape layer is only necessary if you are using shape layer animators or you need to animate the actual vector path, on rare occasions, you may need to animate a fill color or add a gradient to a vector layer, or you may want to use the C4D rendering engine and extrude the layer. Those are the only times you ever need to convert a vector layer to a shape layer. On my last 40 or so AI to AE projects I may have converted seven or eight layers to vector layers, and those layers were specifically prepared in Illustrator taking into account all of the limitations and understanding that only that specific layer needed to be converted to a shape layer. Make sure you are not wasting time. If you have a complex AI file and you convert dozens of layers to shape layers unnecessarily, you could dramatically increase your render time with no increase in quality. 

Almost all of my AI converted to shape layers in the last couple of years have involved creating one or two simple stroke layers that I would use with Trim Paths to draw on some text like animating a signature. That's about it. 

There is no automation for this in After Effects but you have some options out there. You can use OCD Renamer (a 10$ script from aescripts.com) that lets you to rename a group of layers or even the free DUIK utility have some options to rename layers. Also you can use the amazing Overlord by Battle Axe that lets you convert your Illustrator files into Shape Layers from Illustrator retaininng their original names. I think there are other options but all of them are utilities or scripts for AE, no AE native feature to do this.