I've seen historical blogs and articles refer to the holy roman empire as 'the german state', despite it also being the residence of czechs and austrians, and dutch for awhile (when you view it with a modern lens). To me, it seems more like it was the 'austrian state', since the dynasty that it was dissolved under went on to rule austria for another 100 years.
Anyway, the only justifications I've seen for reference to germany are to do with:
How romans referred to hundreds of different tribes for the sake of probable convenience.
The use of the title 'king of the germans' by the pope, who probably got that idea from the romans, intending 'german' to mean 'barbarian' or 'uncivilized'. Hence, a derogatory title.
Term for the language group 'germanic'.
The dutch referred to themselves as 'german' at one point. I have no clue how that trend might have started. I don't think they were instincitvely instilled with that identity at birth (to argue such would be like saying nationalism existed since the stone ages). The same goes for 'teutonic knights' and other such names for various groups within society.
to argue such would be like saying nationalism existed since the stone ages
Nationalism as a political ideology is modern, but political ethnicity (identification of a people with a polity) is as old as history. There's lots of evidence from medieval and early modern sources is that the Holy Roman Empire was indeed identified as a German state, or one in which Germans were the central people. However as you allude to, there were competing non-ethnic definitions. Often the rulers preferred to emphasise its universal Christian and 'Roman' aspects, as these were superior principles of legitimacy at this time and implied overlordship over other rulers, while those opposing the empire it would emphasise its Germanness, in order to rhetorically bring it down to the level of the other Kingdoms.
So for some examples, we have the 12th century English author John Of Salisbury writing "Who appointed the Germans to be judges over the nations?" (Quis Teutonicos constituit iudices nationum?) in opposition to the universal claims of the emperor Frederick I. (Fuhrmann 1994). At around the same time, Gerald of Wales mentions the 'German Empire' in passing, while complaining about the low position of the English post Norman conquest (Coulton, 1935: 32).
In the early modern period we have Martin Luther, writing his 1520 letter "To the Christian Nobility of the German Nation" (An den christlichen Adel deutscher Nation), which pretty clearly identifies the Germans as the 'possessors' of the Holy Roman Empire: "Now, although the Pope has violently and unjustly robbed the true emperor of the Roman empire, or its name, and has given it to us Germans, yet it is certain that God has used the Pope's wickedness to give the German nation this empire and to raise up a new Roman empire, that exists now, after the fall of the old empire." At the same time we also have German scholar Ulrich von Hutten writing to emperor Charles V "For why should Germany have merited to perish with thee instead of for thee?" (Kohn 1961, p. 146)
So to conclude, identification of the Holy Roman Empire with Germany wasn't uncommon in the medieval and early modern periods, whereas I've never seen any sources referring to the empire by 'Bohemia' for example.
References
Coulton, G. (1935). II. Nationalism In The Middle Ages. Cambridge Historical Journal, 5(01), pp.15-40.
Fuhrmann, H, (1994) Quis Teutonicos constituit iudices nationum? The Trouble with Henry. Speculum, Vol. 69, No. 2 (Apr., 1994), pp. 344-358
Heater, D. (1996). World citizenship and government. New York, N.Y.: St. Martin's Press.
Kohn, Hans. (1961). The Idea Of Nationalism: A Study In Its Origins And Background. Transaction Publishers