Yes. We have four Cases in German. The kind how a sentence is build with the articles give a link to the content. For the Nominativ you could ask Who or What is...? ; For the Genitiv you ask for possessive (whose) or because of what what is. ; The Dativ asks "Whom...?" ; And finally the Akkusativ asks "Whom or what?"
Let me give you examples with the dog and the man:
Example: "The dog of the man brought him the mans dog toy."
You can ask: Who or what brought the mans dog toy? (Nominativ) (substantive-question "Wer oder was?")
Because of whose dog toy the dog rans?/ (Genitiv) (possessive-question "Wessen/ Weswegen?")
To whom the dog brought the dog toy? (Dativ) (object/ person-question "Wem?")
Who or what brought the dog to him? (Akkusativ) (object/ person-question "Wen oder was?")
You see that the awsner to all these questions is included in the sentence. That is why we have different articles in German. If the order would be changed but the articles are connected the sentence would still have the same content and keeps clear:
Der Hund gab dem Mann den Ball = Dem Mann gab der Hund den Ball
So back to the: "A man and a dog":
Here is no Akkusativ or Dativ so there is no deeper relationship shown. You have only a man and you have a dog (ein Mann und ein Hund). Otherwise you could ask for example with the Akkusativ-question: Whom or what has the man? - A dog (in this case "ein" changes because of a shown relationship to "einen")
It's got to do with something called case. It's a bit of an abstract grammatical concept, and can take a while to understand. We say that ein Hund is in nominative case and that einen Hund is in accusative case.
Case also exists in English, but has less of an influence than it does in German. For example:
"I see him"
"He sees me"
Notice that the word changes depending on who is doing the seeing and who is being seen. This is analogous to German nominative case and accusative case.
Duo's example sentence has no action involved in it, though. The whole thing is an expression in nominative case.
In German, many words are changed depending on which case they're in. The word ein is one such word. It's also changed depending on gender (e.g. for a female noun it would be eine). Combining all the possibilities of case and gender produces a table, which you should aim to memorise eventually. Looking at that table, we see that einen Hund is accusative case (because Hund is a masculine noun). E.g. Der Mann sieht einen Hund ("The man sees a dog").