Well, the last one was sprite/bsp based. You can emulate the texture style pretty easily by pixelating a relatively low res texture. That may have been a 'duh' response, though.
Here is a quick example, it could be made better with a little work.
The image on the left was a basic tiling wall texture. In Photoshop I just applied a mosaic pixelate filter followed by a smart sharpen and got a basic 'old school' texture.
Umm, there's nothing special about it at all. The artists just worked with the engine limitations they had, which is exactly what you said. Memory limitations, image color bitdepth, animated sprites. You can make the exact same artwork today.
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Here is a quick example, it could be made better with a little work.
The image on the left was a basic tiling wall texture. In Photoshop I just applied a mosaic pixelate filter followed by a smart sharpen and got a basic 'old school' texture.