The colorizing website does actually a complex job. It saves the transparency ie.the alpha values stay unchanged, but the new color is an adjustable mix of original color and selected new color.
Photoshop has got a very simple to use method in another answer. Paint.NET hasn't layer styles, so the proposed method is not available there.
For Paint.NET there are available several colorizing plugins. I haven't tried all of them, only a few. Those are made for adjusting a selectable color range (=change hue, saturation, brightness or transparency). But anyway, the job can be done with them, too. An example:
A random multicolour image with transparency:
Duplicate the image layer and make the copy 50% grey. Use the curves adjustment in RGB-mode:
If no mixing with the original is needed, there's no need to make a copy. Make the original layer grey.
Colorize the grey with a plugin:
Plugin = TR's ColorizerHMS
Adjust the transparency of the colorized layer to define the mix (not asked, but surprising effects are available by changing the blending mode - well worth to try!)
Finally merge the layers if you want to fix the effect to non-adjustable.