So quick primer on German history:

Germany was a bunch of different territorial states that were loosely joined in a confederation. In 1866, one of the states called Prussia, began to make waves. They defeated Austria, and then France and in 1871 united Germany into a federal state. The other German states still "existed," but they functioned like modern American states do. With a federal government and separate state governments.

This continued past WW1. Although the states were no longer monarchies, but were republics. So you still had Prussia, Bavaria, Saxony, etc. existing, but they were now part of a larger German state.

After WWII the allies identified Prussia, which had a reputation as being very militaristic and full of warmongers, as the source of Germany's militarism and thus they wanted to dissolve the state of Prussia. See my answer here for more details on Prussian militarism.

To give a modern example. Imagine if the USA lost a war and was completely occupied. Now Imagine if many of America's leaders came from Texas. The occupying powers, in order to break the hold of the Texans over American politics, dissolved the state of Texas and give its territory to a bunch of different states.

That's what happened to Prussia. The allies figured that if the state of Prussia no longer existed, Prussians would stop being the dominant force in Germany and thus Germany would stop being militaristic and constantly invading its neighbours (which is what the allies assumed Prussians were like).

Is Prussia never allowed to exist again?

Nope, its been officially dissolved and its territory now belongs to a bunch of different German states. And a bunch of Prussia's eastern territory, including East Prussia, is now part of Poland/Russia/Lithuania.

Are people not allowed to identify as Prussian?

You are allowed to identify as Prussian. And certainly some Germans maintained a connection to the state of Prussia for a while after WWII. But keep in mind Germany's culture changed very much in the post-WWII era. And Prussia has been gone for 70+ years. So its more a question of who would?