Argument by equal reasoning

What is an argument by 'equal reasoning' (Latin: pari ratione, sometimes anglo-fied as 'by parity of reason/ing')? Is it the same as an argument from analogy? In his System of Logic (Book III Chapter II "Of Inductions Improperly So Called"), J.S. Mill characterises it as follows (my emphasis):

Having shown that the three angles of the triangle ABC are together equal to two
right angles, we conclude that this is true of every other triangle, not because
it is true of ABC, but for the same reason which proved it to be true of ABC. If
this were to be called Induction, an appropriate name for it would be, induction
by parity of reasoning . But the term can not properly belong to it; the
characteristic quality of Induction is wanting, since the truth obtained, though
really general, is not believed on the evidence of particular instances. We do
not conclude that all triangles have the property because some triangles have,
but from the ulterior demonstrative evidence which was the ground of our
conviction in the particular instances.

This appears very similar to what Kit Fine calls 'argument by arbitrary object'- see here e.g. Take any A. It can be demonstrated that this particular A is a B. But this was any A. Therefore every A is a B.

A similar method is commonly used to show that an argument is invalid. My opponent and I agree that A is true, but he believes that B follows from A. I object that if B followed from A, then 'by equal reasoning' D would follow from C. But we both agree that while C is true, D is not. Therefore B cannot follow from A. My argument depends on the logical form of the argument from A to B, and from C to D, as being essentially the same. Woods and Hudak explain it well*. Although two different arguments differ substantially in their surface structure, they may possess a common 'deep structure', so that one could not consistently hold that one argument is valid, and the other not. Thus:

1. A possesses a deep structure whose logical form provides that the premisses of A bear relation R to its conclusion
2. Argument B shares with A the same deep structure
3. There, B possesses a deep structure whose logical form provides that its premisses likewise bear R to its conclusion.
4. Hence, B is an analogue of A. A and B are good or bad arguments, by parity of reasoning, so-called.

But Woods and Husak also call this an "analogical argument". Is that right? And is the same as an argument I had with a vegetarian years ago? She argued as follows

P1 It is wrong to kill animals for use by human beings
P2 Eating animals is an example of killing animals for use by human beings,
C It is wrong to eat animals.

I pointed out that the following argument was also valid

P1 It is wrong to kill animals for use by human beings
P2 Wearing leather shoes is an example of killing animals for use by human beings,
C It is wrong to wear leather shoes.

Which caused some difficulty for her, as she was fond of (very expensive) leather court shoes. She first denied that the cases really were the same, then ended up modifying the major premiss to 'it is wrong to directly cause the death of animals for use by human beings', arguing that the manufacture of leather is a only by-product of the food industry, and not a direct cause of their death. But this is not an issue about the validity of an argument, but rather its soundness. An argument is unsound if one of its premisses is false. The vegetarian conceded the identity of logical form, and defended the validity of the argument. But it was the minor premiss where the problem lay. Wearing leather shoes is not an example of killing animals for a certain purpose.

According to Juthe, they are sometimes presented as inductive probable arguments. He cites Copi and Burgess Jackson 1992**. Juthe denies this, saying that a true argument by analogy is where the inference goes via an 'analogical relation'. He cites the 'classic case' of Mill's argument for other minds: other humans have bodies like me, act in the same way as me, in cases which from my experience have the same causes, namely feelings and thoughts which only I can observe. Therefore other humans have feelings and thoughts. But this is clearly not the same as argument by equal reasoning.

It seems there is some confusion about the precise distinction between argument by analogy, and argument by 'equal reasoning'.

* "By Parity of Reasoning", *Informal Logic XI.3, 1989
** Informal Logic, Macmillan 1982