MATHEMATICS IS AN ANALOGY.
Mathematical analogy is the safest, and the prefered analogy for physics. It is quite free of emotional baggage, and it is adaptable to any situation that might arise. To be fair, we must note its dangers.
Mathematical equations suggest or imply a false sense of precision and accuracy. The equation for a law or principle, presented without disclaimers and qualifications, fails to convey the limitations of scope and precision that apply.
Equations alone don’t distinguish quantities that are directly measurable and observable, such as mass, length and time, and those that are useful invented concepts, such as energy, momentum, field strength, wave function, etc.
Students often fail to recognize “unphysical” consequences of an equation. Any of the equations for fields such as inverse square fields “blow up” to infinite values at r = 0. Any limit taken by calculus methods obviously can’t be taken all the way to zero values of dx or dt at the quantum level.
Mathematical expressions of fields and wave functions tempt students to think of these things as having a “real” existence on a par with macroscopic quantities like mass or distance.
These misconceptions are not the fault of mathematics, nor the fault of physicists. Physicists are fully aware of these facts, and in a full treatment of theory they are all dealt with properly. It is only when the physics is diluted, simplified or dummied down for popular consumption, that important qualifications and disclaimers are left on the cutting-room floor.