First of all make sure that the object is UV unwrapped so that the image texture can be shown correctly on the object - but if you do not know how to unwrap, this would be the topic for another question.
Let us assume the model is UV unwrapped. To show the image texture in the Viewport Shading mode Texture, it is necessary that the object has a material on it which contains one or more Image Texture nodes.
In the following example, the object has a material with three texture nodes: two Image Texture nodes, one with a color grid and another with a grey uv grid and the third is a Noise Texture. Only the color grid is connected to the Base Color of the Principled BSDF.
If you now select the Image Texture node with the color grid or that node was the last texture node selected (so it doesn't matter if you have deselected everything afterwards or maybe selected the BSDF node) then the viewport will show the color grid texture:
If you now select the other Image Texture node with the grey UV grid, the viewport will display this one (and still show it after deselecting). It does not matter that the Image Texture node is not connected to anything - this mode is for previewing what you select, so to see different images on the object you do not really have to switch the node connections:
It is no problem to select the Principled BSDF or the Material Output node or others after selecting the Image Texture node, the image will still be showing. However, if you select another texture node that is not an image but procedural, like Noise Texture, Voronoi Texture, Gradient Texture etc., these textures will not be displayed in the viewport. Instead the object will show the Color which is set in the material under Viewport Display (by default white or light grey rather, in this case green to clearly show it):
The same happens with materials that do not have any Image Texture at all. In this case there is no image which could be shown in Texture shading mode, so the Viewport Display > Color will be visible: