No, it's not. A castle is a fortified place of residence, with heavy emphasis on the military needs in its architectural designs.
Neuschwanstein was built in the 1880s, at which point any fortification that could have fulfilled its military role you would not want to live in, thanks to improvements in artillery. Castles are functionally obsolete around the 15th century, when gunpowder-based artillery became common and you needed substantial more earthworks and enveloping lines of fire rather than the straight curtain wall you see with castles.
Neuschwanstein is instead a 19th-century version of the McMansion: it's a house (palace, really), built to (then-) modern standards that is dressed up to look like some sort of fancy architecture, in this case, a Medieval castle based not on historical research but in the (then-) modern pop culture version of it.
It's a fake castle inasmuch as it was not built out of some military defensive necessity, but a hunting lodge for the rich and powerful to enjoy a retreat. So basically a fancy vacation home. But of course it's a real place with real historic value and absolutely worth a visit.
