You can now, with Adobe ® Photoshop ® CC CS6, easily change the color of the shape via the fill and stroke settings dropdown along the top bar of Adobe ® Photoshop ®. You can also fill them with gradients as well as patterns and also apply a variety of different stroke settings to the art (though sadly not including the beautiful strokes from Illustrator - perhaps for another version ?). They are all accessed via the custom shapes tool and the preset's panel. You can also fill them with images via the patterns as you can define any large image as a pattern and then select that from the dropdown panel. Below, the vector artwork (as a layer) can be re-colored while the CSH tool has been selected. The layer fill color can be set to a solid color as well as gradient and pattern. If you need a large selection of amazing Photoshop custom shapes then you can find 100,000 of them on this site
You can also set the fill to "No color" and I know that sounds a little boring but actually it is very useful as you can then just use the stroke (both stroke and fill can be no fill and that can have their uses such as layers with a shadow and other effects). The no fill means you can create all kinds of artworks only with strokes.
You can set the fill to a solid color for the artwork layer. You can use any of the colors from the swatches displayed as well as add any swatches sets available.
You can set the color to a gradient, select any of the items included in the palette. You can 1000s of additional presets such as our Photoshop gradients collection but there are many additional ones available on the web. You can load them via the right side menu such as replace and load. You can set the angle and mode etc though you are limited in the things you can change via the panel (you can use the next technique to alter the origin etc). If you have a single vector artwork (this does not appear with more than one selected) then you can also alter the gradient via the layer menu and layer content options so you can change the gradient to a radial or diamond as well as change the origin as well as scale and more
You can set the color to a pattern. You can use any of the pre-defined patterns included in the panel as well as any of the items available via the right side menu. You can also define your own or use ones ready made available on the web (such as on this site). You can also use any super large image and then define that as a pattern and use that via that option (if it is much larger than the shape then it doesn't matter if the artwork is seamless or not). You can also set the scale for the artwork. If you have only one piece of artwork selected then you can modify the pattern such as origin and scale etc via the layer menu layer content options.
You cannot change the fill color via the dropdown if you select the pixel option. You can change the color before you add the artwork. A quick workaround for this is to add the vector to a layer and then use the various adjustment functions to change the color at a later date. You can also the magic wand to select the area of the custom shape in Adobe ® Photoshop ® and this can then be filled with gradients, images (using paste into command) as well as patterns (via the layer fill content)
You can also just copy the artwork to Illustrator and re-color it there. You can turn the artwork into a smart object and then use smart adjustments to re-color it such as black and white or LUT etc You can also filters such as camera raw to modify the color etc. Certain there are even more ways
Letter custom shapes have been used on this page