The fallacy of a false analogy happens when someone is making an argument in the form of an analogy where the analogy is lacking in certain aspects that make up a good analogy.
The basic form of an argument from analogy tends to be similar to this:
P and Q are similar in A, B, and C
In P we have also observed X
Therefore Q also probably has X
An example of such an argument could be
Ivan and Boris both work out hard, eat right and get extreme results in body composition
We also see that Boris takes anabolic steroids.
Therefore, Ivan probably also takes anabolic steroids.
The factors that either add to or detract from an argument from analogy are:
A) Relevance (positive or negative) of known similarities of the two things to the similarity inferred in the conclusion.
B) Degree of relative similarity or dissimilarity of the two things.
C) The amount and variety of instances that form the basis of an analogy.
This is generally not the type of fallacy you end up evaluating on the fly. In verbal discourse analogies tend to be superficial at best, and used for humorous effect, rather than as a piece of complex reasoning. They serve a purpose in rhetoric by “short-circuiting” a persons mind, by triggering associations.
For instance, when I say that “My opponent is like Stalin” the audience’s minds start doing the X from the argument themselves. I don’t even have to do the A, B, and C for this to happen. This is why good analogies are central to becoming a great orator or manipulator. I remember hearing someone say that the worst way you can punish a child, is by telling them “Go to our room, I’ll be up to punish you in 10 minutes” because the child’s mind starts creating their worst case scenarios for what the punishment will be.
This is what a great orator does with the false analogy, he creates what appears to be an argument, that the audience then convince themselves of in their mind.
Speaker: Trump is like Hitler!
Audience: He said it, so in what ways is Trump like Hitler?
What happens in their mind is that they come up with the best reasons for why Trump is like Hitler, rather than evaluating and dismissing it as fiery rhetoric. This is also congruent with salesmanship tactics, where putting your client in the position where they are selling your product to themselves increases the chance of making a sale.