I'd say it's definitely a data structure, but one with a canonical string representation. It may not be fundamentally a string, but that representation is critical to how we convey it's meaning to such extent that it's used across languages and browsers as the method of transport.
JSON is conceptually simpler and there's still a lot of quirks between libraries for that already (e.g. null/undefined, integer/decimal representation, and large numbers). XML has more going on to start, and then you get all the different libraries inventing their own abstractions as you said and it picks up a lot of pitfalls. FWIW; I've messed around a lot with configuration languages and it's definitely hard to get right so I understand how this differences accumulate.
The best way to represent HTML is with HTML itself, not an HTML-like system that varies in subtle ways between languages and libraries that happens to be missing the one feature you need.
I'm a large fan of Embedded DSLs (Domain-Specific Languages) for these types of problems, as they allow using normal HTML syntax directly in the rest of your code. Combined with macros (for compile-time parsing/analysis) and interpolation (for safely templating values), DSLs can be safer, more performant, and overall more maintainable than other approaches. For languages like HTML and SQL that are standard and well-defined, this is undoubtedly a better approach in my mind.
I've been developing my own approach to Embedded DSLs as my thesis research with my own programming language Rhovas [0], which addresses concerns with syntax restrictions (e.g. string quotes), semantic analysis (AST/IR transformations), and most importantly tooling. Happy to answer any questions about work in this area.
JSON is conceptually simpler and there's still a lot of quirks between libraries for that already (e.g. null/undefined, integer/decimal representation, and large numbers). XML has more going on to start, and then you get all the different libraries inventing their own abstractions as you said and it picks up a lot of pitfalls. FWIW; I've messed around a lot with configuration languages and it's definitely hard to get right so I understand how this differences accumulate.