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The texture can be saved with a custom brush but a color can not.

All Photoshop brushes (used with any tool) whether custom or otherwise, are made using a percentage variation of black to white with no color information. Using the Brush Tool, the color of the brush is determined by the foreground color setting.

A common misunderstanding is with the "Color Dynamics" setting in the Brushes Panel. This only determines how the chosen foreground color will behave during the brushes use.

Your only option is to add the preferred color to your color swatches panel and select it as your foreground color when you use a particular brush.