LOT of mis-information and intentional misleading info about jewelry ...
It is true that, in general, metals don't "play well" with living tissue, metals react chemically (corrode) and create conditions for all sorts of problems. Some metals are highly toxic or cause cancer. And... this is not unique to metals, humans have created all sorts of chemicals that don't exist in nature (or, at least did not until we started polluting the world with them).
We use very few high purity metals for anything, much less for jewelry so, in general, the metals you come in contact with are alloys (combinations of a metal element with one or more other element).
But... all this said, there are metals developed specifically to be compatible with the human body. Metallurgists have developed, for example, alloys for implantation inside the human body for supporting broken bones, replacing joints, for devices like pace makers and insulin pumps.
On your specific problem... Yes stainless steels are alloys. In general, stainless steels are Fe-Cr or Fe-Cr-Ni alloys with small additions of carbon. There are over a hundred different commercial stainless steels for many different applications. I am not sure which specific stainless alloys this includes but some stainless steel is specifically designed to work well with the human body.
I doubt you are really allergic to gold. Gold is about the most inert metal we have. What you are probably allergic to is the alloying elements that are added to gold to make jewelry. In jewelry, "pure" gold is 24 carat but it is so soft (and so expensive) that almost no one makes pure gold jewelry. Common jewelry alloys are 18 carat (18/24ths or 75% Au) or 12 carat (12/24ths or 50% Au). The other "stuff" in the gold varies but often includes Ag, Cu, sometimes Ni, and small amounts of other elements. Cu and Ni in particular are known to cause skin problems. And... even if it is "pure gold plated", the gold plating is probably so thin (MUCH thinner than a human hair) it wears off almost immediately and your skin is in contact with whatever alloy is underneath.
It is often true that you get what you pay for. So, if you buy cheap jewelry, you get cheap jewelry alloys.