Can anyone identify the item in the photos? It is in the circuit with an electromagnet on an old foot massager.

 

It is called a selenium rectifier.

 

Those selenium rectifiers, when they failed, would release an acrid smoke.

 

Those selenium rectifiers, when they failed, would release an acrid smoke.

Acrid sounds just too clinical given the actual experience of the stink that comes from a burning selenium rectifier. While there are some things that are unpleasant but worth experiencing to be well rounded (e.g.: a powerful but non-deadly shock due to carelessness, an exploding electrolytic cap in an HV supply… because of carelessness, etc.) selenium stank is not something anyone needs to experience first hand.

Ugh, it is rank.

 

When they burn, Selenium Dioxide is often produced. If you have them on actively running gear, I would replace them.
https://www.finarchemicals.com/msds/Selenium dioxide.pdf

 

What is the power supply part of?

I would consider 1N5408 (3A) or 1N4007 (1A) for a silicon replacement, plus a dropping resistor because selenium diodes have a higher internal resistance- which some devices rely on. They also don't mind spikes and overload, so extra parts can be required to moderate that with silicon parts.

Selenium rectifier diodes were used in aircraft i.e. B-52 in 1957 and we can all imagine the toxic smoke when they failed.
They are also attacked by fungus, and will simply die of old age. Some customers still want them for cathodic protection rectifiers because they tolerate the (ground) lightning strikes.

 

Those selenium rectifiers, when they failed, would release an acrid smoke.

When my nose detected a failure of one, it smelled like cat's pee.
The one shown appears to be 1/2 wave.

 

I still have the transformer for the train set my brother and I had as kids. It contained one of those selenium rectifiers to deliver ~12VDC output.