Try to pass the third parameter as string which refers property name, `filePath` here.

    self.wizardPage.registerField('projPath*', self.projPath, 'filePath', self.projPath.fileChanged)

Here is a working example in QGIS Python Editor:

    from qgis.PyQt.QtCore import *
    from qgis.PyQt.QtGui import *
    
    
    class Wizard(QWizard):
        def __init__(self, parent=None):
            super(Wizard, self).__init__(parent)
            self.addPage(Page1(self))
            self.addPage(Page2(self))
            self.setWindowTitle("Test")
    
    
    class Page1(QWizardPage):
        def __init__(self, parent=None):
            super(Page1, self).__init__(parent)
    
            self.projPath = QgsFileWidget(self)
            self.registerField('projPath*', self.projPath, 'filePath', self.projPath.fileChanged)
            self.projPath.fileChanged.connect(self.file_changed)
            
            layout = QVBoxLayout()
            layout.addWidget(self.projPath)
            self.setLayout(layout)
            
        def file_changed(self):
            self.setField("projPath", self.projPath.filePath())
        
        def isComplete(self): # Re-implementation of isComplete
            return bool(self.projPath.filePath())
    
    
    class Page2(QWizardPage):
        def __init__(self, parent=None):
            super(Page2, self).__init__(parent)
            self.label = QLabel()
            layout = QVBoxLayout()
            layout.addWidget(self.label)
            self.setLayout(layout)
        
        def initializePage(self):
            path = self.field("projPath")
            self.label.setText(f"Path: '{path}'")
            
    
    wizard = Wizard()
    wizard.show()

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/mljZJ.gif