One of the key features of graph databases is to select one node, and then recursively 'chase' edges until you find another node matching some criteria. Other database models can have trouble representing chasing an unbounded number of edges. E.g. in the relational model, following an edge to another node is usually represented as a Join operation, and SQL doesn't let you parameterize the number of iterated joins. This is especially true if the thing you want to query is actually the path length.
In an HN thread from a few days ago, someone made the claim that the graph model could be represented by SQL + recursion, and recursive SQL is an extension offered by some databases. But the relational model itself cannot fully represent the graph model.
Without digging too deep, I suspect other database models run into similar problems. E.g. a document store could very easily represent a Directed Acyclic Graph as a document, but when you get into general graphs your document needs to end on a value that is the key to another graph.
This is not agree with the claim that graph databases are generally superior. I like them, and they're fun, and I think more developers should be aware of them for cases where they apply, but I also don't think they have advantages over relational or document stores when the data is natively table-shaped or DAG-shaped.