= Two templates in one, told apart by whether they are given an argument.
Bare, {{eq}} is an equals sign — the one character that cannot be written literally inside a template call, an = there being read as naming a parameter. That is all this template was until 20 August 2026, and it still is: $x_1{{eq}}x_2$, as in Nogueira02a.
Given one to four Anno Fabri tags, it is a reference to the equations carrying them, and it counts:
{{eq|00fC1}} → Eq. (3){{eq|00fC1|00fD2}} → Eqs. (3) and (4){{eq|00fC1|00fD2|00fE3}} → Eqs. (3), (4) and (5)The numbers are MathJax's, counted afresh at every rendering, so inserting an equation above renumbers every reference to it. That is the whole reason for naming an equation instead of numbering it by hand. Each reference is a [[Template:Eqref|{{eqref}}]], which can also be used on its own where the sentence supplies its own wording.
Since 1 September 2026 the reference goes through Module:Eq, which is what lets it point at an equation on another page: {{eq|05hUD|in=August (2026)}} from September (2026) gives Eq. (1), a link landing on the zeet that holds it. The number there cannot be MathJax's — MathJax numbers one page at a time and knows nothing of August's equations while it renders September's — so it is counted on the server out of August's own wikitext, once, while this page is parsed. The reader fetches nothing; the number cannot go stale either, an edit to August re-rendering every page that points into it.
With no in=, the tag says where to look: it is an Anno Fabri tag, hence an instant, and the month page of that instant is tried, on 𝕐 then on ℤ, and then the month before it — the tag is stamped when the Eq button is pressed, which is not always the month of the page the equation ends up on. Failing all of them it stays the plain \eqref, which is MathJax's way of saying it has never heard of that label.
The tag is the one in the equation's own \label{eq:00fC1}, which the Eq button of the edit toolbar writes (see MediaWiki:Common.js: it turns a $$…$$ into a numbered \begin{equation} and names it after the second it was pressed). Base62 and case-sensitive: 00fC1 and 00fc1 are two different seconds, and a reference spelt in the wrong case points at no equation at all.
The tie between Eq. and the number is a real and not a $\mathrm{\LaTeX}$ ~, which wikitext does not know and would print as a tilde; M-O mends those where they are written by hand. The on its own does not hold, though, and never did: the number is not text but a MathJax box, and a browser will start a line with an atomic inline however the space in front of it is spelt. Measured over sixty line offsets on 30 August 2026, three of them broke Eq. from its number. So the pair is also wrapped in a white-space:nowrap span, which does hold — nought in sixty. Only the word and the first number are tied: the commas and the and of a multiple reference are ordinary places to break a line and are left as they are. Five tags or more is one reference too many for a sentence to carry; write them out.