To be clear, let me use "$=$" to mean the same element in a set, and "$\sim$" to mean neither "$>$" nor "$<$" in an order.
In an ordered set even if $x\ne y$, we can still set $x\sim y$, right?
If yes, is it true that for an ordered field, under the extra conditions, viz.,
$$x+z<y+z \hspace{1cm} \text{if} \hspace{1cm}y<z$$ $$xy>0 \hspace{1cm} \text{if} \hspace{1cm} x>0 \hspace{1cm} \text{and} \hspace{1cm} y>0$$
we can prove that $$x\sim y \hspace{1cm}\text{iff}\hspace{1cm} x=y$$
Generally speaking, in almost all circumstances it is the case that $x = y$ iff $x$ and $y$ are "the same" (semantically) or "completely interchangeable in all circumstances" (operationally). Of course, in your theory you can define equality whatever way you want, but then you should probably use a different symbol since otherwise everybody would be confused.