I usually pin my most accessed websites on start screen. The pinned tiles take the shape of site's favicon which sometimes look ugly. Is there an app which let me change the default site tile from a list of custom icons.
No. There isn't any app for that. The tile itself is created in a such way that it would pick the up the image whatever the site has and will assign it as tile image.
For example if you pin the stack exchange site to start screen. The default stack exchange logo will appear on tile.