But what is the standard?
The standard is not being different at all cost, being different is the question not the standard.
The standard concerns on what basis a shape is different from another.
So the question becomes: on what basis does one shape stand out?
If I say: now it is unique because it lacks a unique feature other shapes do have, I cannot say that this uniqueness
is ex- or interchangable with the unique features of the other shapes. I cannot exchange the basis.
One deals with a state of being, the other with having a well defined property.
And in that state of being, lacking the unique property, it stands out. To be or to have, that is the question.

It seems to me that both the puzzle and the Monty Python scene (see link) are examples or Russell’s Paradox, which asks if a set which is not a member of itself is a member of the set of all sets that are not members of themselves. If it is, it isn’t, and if it isn’t then it is.

Similarly in the Monty Python scene the crowd on being told that they are “all different” look at one another and proudly respond in unison “yes we are all different”, all except for one who has to report that he isn’t. As soon as he does so he becomes different, as the only one who isn’t different, but then immediately becomes one of the crowd who are all different, and so on …

With the shape puzzle (and “odd man out” puzzles in general) we are looking for some set of features that all the shapes share, except for one. So, on the first pass, we see that common features, shared by all but one are: colour (red), shape (square), size (big), and border (present). But now we have a problem, there are four odd ones, not one, so we need to look for another feature shared by all but one of the shapes, so we add a fith feature, to match the “standard” the shape must have exactly one “non-standard” feature. Now (counting from left to right) shapes 2 to 5 can all proudly proclaim, yes we are all different, but poor old shape number 1, with its boring large square shape, plebian red colour, and common as muck thick black border, must nervously half raise its hand and say “I’m not!” But being different is part of the standard, so it has one non-standard feature, so it doesn’t, so it does, and so on …

But wait, if it is not different, it has exactly one feature that is not part of the standard, so it does match the standard, but if it does match the standard it is not different to the other four shapes, so does not have one differing feature, so it does not match the standard, and so on …

So the question is, if a set is defined as being those objects that meet a certain standard except in one respect

In the second puzzle you see that you will always need two steps the get from one figure to the other.

Or you can say that it is the first because this form has no unique feature on its own.
This is what I saw.

But if you talk about the one that is ‘most unlike’ the others, then it is definitely the green one - it jumps at you the moment you see the question.
But this way of looking at it is not analytical.

This question has an obvious analytical pattern here however: it is deliberately created in order to make the one that is not odd stand out with a gameplay of size, colour and shape.
That is the essential meaning of this question, and only one answer addresses that.

Kinship in forms, as if they evolved from each other, is another perception.
Here also the first form takes central stage: there is a dwarf ’subspecies’, a green and a round variety, and the variety without a shell / exoskeleton.
In that case it is not usual to say that the central, holotype, form is the odd one out.
It is the one that STANDS out from the others because its form is typical for ‘the species’.