The solid mode is not meant to display textures or materials. It displays the textures just in texture paint mode, with a flat shading. So that you can paint your textures.
In object mode you need the Material Preview mode to display textures. But this mode is, as the name says, to preview material and textures. Although it can calculate shadow, it does not calculate global illumination etc. . When you want to see in the viewport what you get in the rendered image, then you need to use the viewport shading called rendered . That’s the button at the right. This mode also works in texture painting.
But even this display may show a difference to your final rendering. There is a reason that you need the final rendering …