It depends on how you look at it, I think. I experience Anglo-Saxon, Middle Dutch, Modern German and Modern Dutch as representatives of the same language, for instance. Despite the fact that they are not mutually intelligible, there is a lot you recognise if you know one of these languages and then begin to study one of the others. To illustrate what I mean I've translated the first sentence of the Dutch romance of 'Karel ende Elegast' into Moder Dutch, Modern German en Modern English below. Modern English, of course, is interesting as its largely romance vocabulary sets it somewhat apart from the other Germanic languages.
Fraaie historie ende al waer mag ik u tellen; hoort naer.
Een fraaie geschienis, en helemaal waar, mag ik u vertellen; luistert nader.
Eine schöne Geschichte, und ganz wahr, kann ich Ihnen erzählen; höret andächtig zu.
A beautiful story and entirely true I may tell you; listen carefully.
Studying Latin, there is a lot I recognise from the lessons in French and Spanish that I took in secondary school. Obviously, a lot has changed, but the paradigm of Spanish 'hablar' is still strongly reminiscent of the Latin first conjugation (hablo, hablas, habla, hablamos, habláis, hablan...) and I now finally understand where the French word 'quotidien' comes from.
You "experience" Anglo-Saxon, modern German, and modern Dutch as the same language? The differences are tremendous, especially in the case of Dutch, whose adjective declension has been simplified a hundredfold. Vocabulary is far from everything in a language, and even there, though you rightly point out the amount of Latinate vocabulary in modern English, you miss the fact that Dutch itself borrows far more extensively from Latin than German does.
I think the German should say:
Eine schöne Geschichte, und eine ganz wahre, kann ich Ihnen erzählen; hört andächtig zu.
The undeclined adjective ("wahr") would give it adverbial force. Of course, this awkwardness could be avoided by using what is more natural German word order:
Eine schöne und ganz wahre Geschichte...
You've really stretched the English to fit the Dutch, and the result is exceptionally awkward and bordering on ungrammatical. Along with the adoption of Romance vocabulary, English also adopted Romance word order, and we're not so free to put whatever we want at the beginning of the sentence as other Germanic speakers are, nor are we obligated to ensure the verb is placed second, nor do we have to put infinitives and participles at the end, nor can we pile lots of elements onto an attributive modifier before a noun (
diese von meinem guten Freund erzählte Geschichte) etc... Using the word choice of your sentence:
"I can tell you a beautiful story, and an entirely true one (to boot/ as well/ no less); listen carefully."
If you want to invert the word order to emphasize the story, you have to grammatically restructure the sentence, unlike in other Germanic languages:
"It's a beautiful story I'm about to tell you..."
"This is a beautiful story I'm about to tell you..."
"I've got a beautiful story I can tell you..."
"There's a beautiful story I can tell you..."
Dutch itself has lost cases almost altogether, which certainly distinguishes it from German (the same has happened in the Romance languages, although there are still five cases in Romanian (though morphologically, there are only two). English has lost grammatical gender altogether; except for Romanian (afaik) (and a couple of pronouns in Spanish), the Romance languages have lost the neuter.
Even in the vocabulary department, your example doesn't do too well, though admittedly, most of the words in the Dutch are related to words in English and German, just not the ones you happened to need here.
Though the verb paradigms in the Romance languages tend to show similarities to those of Latin, the wholesale replacement of the Latin future with infinitive + habere (
tenebo -> *tenere habeo) is a feature of medieval Latin, and certainly distinguishes modern Romance from Ciceronian Latin, as is the formation of the conditional (a nonexistent tense in Latin) from the imperfect of the above construction (
*tenere habebam). Even more markedly distinct is the extinction of the Latin perfect in some Romance languages (if you consider the
passé simple and
passato remoto extinct), which, by the way, mirrors the disappearance of the
Präteritum from spoken German, though somewhat more pronouncedly. In fact, the compound perfect and compound passive are both elements of Romance that aren't found in Latin (except in the passive perfect tenses, which are "doubly past" in Romance, with both a past participle
and a past-tense verb).
In short, the differences between the various Germanic languages are enormous, in grammar particularly, but also in vocabulary; the differences between Romance and Latin are even greater in grammar
and in vocabulary. To "experience" English, Dutch and German "as the same language" is rather strange given the differences between them, and to "experience" French, Spanish, and Latin as the same language is equally if not more strange. While I understand where you're coming from, I hope I've shown you that it isn't just by convention that we refer to these as different languages.