I know that the Baltic nations were briefly part of the HRE, but not for long, and that would have been a long time before this image was made. So why would the author have seen the Baltic nations as Germanic? Was this a common belief?
First time answering a question here, I'll do my best to keep things succinct!
Essentially there are two aspects to this - one is the process of Ostsiedlung, and the other the history of crusading states in the Baltics.
During the early medieval period (in the 11th-15th century) a large number of German merchants and settlers came to the Baltic and Poland. What's important to note is that this wasn't the result of any concentrated effort of "Germanization", but rather a long process driven by trade and economics. In any result, there was a notable population of Baltic Germans all the way until the second world war.
Secondly is the history of the Baltic Crusades. Many popes called for crusades in the Baltics to bring Christianity to the region and wipe out local pagans, which various holy orders used to establish bases of power in the Baltics. The most well known of these is the Teutonic Order, which features in a lot of national myths of Poland as this evil, all-encroaching enemy, which later became the Duchy of Prussia through a lot of weird succession stuff.
Obviously the reality of these states and population movements is nuanced, but they became symbols of nascent German nationalism in the 19th century. With the unification of Germany, there was a huge rise in efforts to create a cohesive national identity and shared legacy - in the art you posted, you can see Germans represented by classical depictions of Roman-era Germanic tribes, a common target for budding nationalism even though their connection to what is commonly understood as "Germany" is, well, complicated. The history of Germans in the Baltics became combined with 19th century ideas of colonialism and imperialism, reinterpreting these early trends in the context of imperialism and aggressive nationhood. Thus, while the Baltics were never part of "Germany proper", they became a common target of irredentist propaganda and imagery, started by 19th century imperialist rhetoric and, well, driven into high gear by the rise of Fascism.
TL;DR: various economic, political and religious movements led to a German population in the Baltics over the medieval period, which was then used to justify imperialist claims. This kind of mythologization and embellishment of historic trends for political purposes was all the rage in the 19th century,
