"We understand you."
Translation:Wir verstehen Sie.
Why is the "you" in "we thank you" an indirect object (Ihnen), while the "you" in "we understand you" is a direct object (Sie)?
No idea. danken is simply one of a group of verbs that takes its object in the dative case, rather than accusative case -- just something to learn.
helfen, danken, folgen, gefallen, gehören, antworten are some of the more common verbs that take the dative case for their object.
From another Duolingo post, with some modification:
There are many ways to translate "you" into German:
du (accusative dich, dative dir, possessive dein) is used in the singular informal to speak to a single person whom you are familiar with (you should avoid using it with strangers and people with authority). It corresponds to the French "tu".
ihr (accusative & dative euch, possessive euer) is used in the plural informal to speak to multiple people whom you are familiar with. It is the plural equivalent of "du".
sie (accusative sie, dative ihnen, possessive ihr) is the third person plural pronoun, corresponding to the English they
Sie (accusative Sie, dative Ihnen, possessive Ihr) is used in the singular and plural formal to speak to one or more people in a formal manner, whom you would not use du or ihr with. It is only distinguished from the third person plural pronoun "sie" through capitalisation.
Yes, there are specific pronouns for each gender, but only in 3rd person sing. and pl. In 2nd person there are basicly sing. "du" and pl. "ihr" regatdless of which gender the adressed person has. However, in German exists also a Formal You which is "Sie" both for sing. and pl.
Does this sentance use Sie to reffer to the formal way to say you (singular or plural)?
Would "Wir Verstehen Ihnen" be used to refer to the plural of "you" in the informal context?
No. That would be Wir verstehen euch.
Ihnen is the dative case of formal Sie. What you need is the accusative of informal plural ihr, which is euch.