Looking at the docs again with fresh eyes, I think you and fulafel are on the money.
The specific engine syntaxes are by & large mutually incompatible, but DOT does seem to be the label used for the overall lang as well as the dot-engine-compatible dialect.
Various tools use tool-specific graph attributes. For example, "rank" and "minlen" mean something to the hierarchical or layered graph layout tool (dot) but not to other layout tools. "size" and "label" are the same in all the layout tools. They all use the same underlying graph representation library with a parser generated by yacc or bison.
The documentation includes a big table of attributes that graphviz tools recognize.
With the availability of LLMs, there is better automated support now to find features that are needed. Just imagining here, but "make the layout fill the available space" or "make all the nodes look like points with associated text labels" (not sure if that even works but it should).