(Note: original question asked about using wood stain on stainless steel.)
I doubt a wood stain would adhere to metal of any sort whatsoever, unless the metal was very rough and the stain settled into some of the depressions leaving some behind when the solvent evaporated. Even if it did, the color would be very succeptable to being scratched off because it simply wouldn't adhere in any meaningful way.
My initial thought was to recommend anodizing. However, according to this article*, you can't (effectively) anodize stainless (refer to the article for more details on that).
However, he does recommend, several paragraphs down, a couple of ways of coloring stainless steel:
“Flame Anodizing” or Heat Coloring
This goes by a few different names – temper coloring, heat coloring, heat anodizing, flame coloring… you get the idea.
He includes a color/temperature chart to get colors from "pale yellow" (550°F) to "dark blue" (1110°F) for 304 stainless. This may or may not be something you can achieve with accuracy at home.
About this method, he makes this note:
The cons? The coloring scratches and wears off super easy. Like you could softly scrape your car keys across the colored surface and it’d come right off. So if you want this to last somewhat longer, you might want to consider using some kind of clear coat.
He also recommends:
Hard Coating (Ceramic, etc)
This is my preferred way to color stainless steel. There are a lot of options and great products out there...
He suggests, but doesn't highly recommend:
This is the “economy” way of coloring stainless. While there are some expensive paints out there, the large majority are uncomfortably budget friendly with performance to match.
Note that this is different than "staining", especially with a stain designed for wood.
He wraps up with:
Usually this is a process done to regular steel to make it more corrosion resistance, but you can also do it to stainless. This is definitely a possibility if you’re just wanting to make the stainless black.
Quite literally the first response in my search for "anodize stainless steel". I cannot vouch for its accuracy or quality. All facts, figures and quotes taken from that article.