The various states pre 1870 would have had identities different from the Prussian, and with unification happening there would have been need to create a new identity: German. Except what kind of history would that German identity have? Was it prussian history combined with all the other german histories? Did the Prussian receive more attention?
I was thinking of an example: someone like Frederick the Great who was a Prussian king who did great things for the Prussian state. After 1870 did people conceive of Frederick as a German king who did great things for the German people and that's why we find him important? I imagine that could piss off the Prussian because that feels like someone appropriating your own history, something that belongs to you and not to them.
But then if appropriating isn't possible, does that mean every state just continued to teach their own story? Because that seems kind of counterproductive to the process of creating a nation-state.
The 19th century essentially saw the first formation of a "German" identity. What was proposed was heavily romanticized, nationalistic and by modern standards largely a-historical. In many regards it would only be reevaluated and deconstructed fully by historians during the onset of the 1960s.
It's important to note though, that it did not follow the formation of the German Empire; it preceded it by some decades in intellectual and/or bourgeois circles, really catching steam during and following the Napoleonic Wars.
In this nationalistic view, "Germans" had been a people since time immemorial. The various Germanic tribes, had been Germans proper. These tribes had toppled Rome but their unity and hegemony (under Charlemagne) had been disrupted (by the French) which had led to the Kleinstaaterei, which led to national disunity and the "Germans" being nothing but pawns in the hands of other, more unitary European states.
A relatively small group of intellectuals had wanted to start "Germany" from scratch (that is to say, based on the French and American Revolution) as a republic via revolution in 1848, but failed. As you say, the German Empire was created by military force.
Nevertheless, in 1871 Prussia/Bismarck had a lot of nationalistic rhetoric to fall back on. They certainly did not have to create a new German identity. They also did not have to try hard to "sell" a Prussian-centric view of history. Prussia had been (as Brandenburg) one of the electoral states of the Holy Roman Empire, it had been one of the preeminent German states since the 18th century, one of the (eventual) victors of Napoleon and in 1871 had defeated both Austria and France thereby restoring "German greatness"; which gave it a lot of prestige. So yes, to use your example, Frederick the Great became a great German king, as well as a great Prussian king.
(saying all of Prussian history became German history would go too far though; as with many European countries, there's a big foggy gap between the Frankish Empire and the Renaissance/French Revolution in which history was a mix between generalisations and eccentric cherry picking)
In the end, the only thing that the Prussians/Bismarck fought over with regard to German post-1871 identity politics, concerned religion; as he came into conflict with the Catholic Church concerning their influence within the new state. This created a degree of animosity towards Prussia in the southern (and traditionally catholic) states of the German Empire; which in a way lives on today in the form of North-South cultural stereotypes.