Contrary to the comments appended to the question, I think the notion of analogy can be made precise.

Definition: An analogy of concept A defined in setting SA, is a concept B defined in setting SB such that there exists a generalized setting SX which includes both SA and SB as example settings, and such that there also exists a concept X defined in setting SX which reduces to concept A or concept B when attention is restricted to either setting SA or SB.

In general, an analogy is not unique. A concept could have many analogies, and even for a particular analagous analogous concept there could be more than one way in which it is considered to be analagousanalogous.

Example: In Time scale calculus which unifies difference and differential equations, there have been publications with differing answers over how to define the analogy between discrete and continuous transforms. A particular description which encapsulates both the integer and real number transforms may apply to other sets such as the rationals, but a different description might not apply to Q. So an analogy is not just two objects but also the link between them.

Contrary to the comments appended to the question, I think the notion of analogy can be made precise.

Definition: An analogy of concept A defined in setting SA, is a concept B defined in setting SB such that there exists a generalized setting SX which includes both SA and SB as example settings, and such that there also exists a concept X defined in setting SX which reduces to concept A or concept B when attention is restricted to either setting SA or SB.

In general, an analogy is not unique. A concept could have many analogies, and even for a particular analagous concept there could be more than one way in which it is considered to be analagous.

Example: In Time scale calculus which unifies difference and differential equations, there have been publications with differing answers over how to define the analogy between discrete and continuous transforms. A particular description which encapsulates both the integer and real number transforms may apply to other sets such as the rationals, but a different description might not apply to Q. So an analogy is not just two objects but also the link between them.