Как задать разные ширины колонок у QTreeView

Попробуйте заменить строку:

    self.header().setSectionResizeMode(QtWidgets.QHeaderView.Stretch)

на строки:

    self.setColumnWidth(0, 20)
    
    self.header().setSectionResizeMode(1, QtWidgets.QHeaderView.Stretch) 
    self.header().setSectionResizeMode(2, QtWidgets.QHeaderView.Stretch)
    self.header().setSectionResizeMode(3, QtWidgets.QHeaderView.Stretch)
    self.header().setSectionResizeMode(4, QtWidgets.QHeaderView.Stretch)
    self.header().setSectionResizeMode(5, QtWidgets.QHeaderView.Stretch) 
    self.header().setSectionResizeMode(6, QtWidgets.QHeaderView.Stretch)
    self.header().setSectionResizeMode(7, QtWidgets.QHeaderView.Stretch)
    
    self.setFocusPolicy(QtCore.Qt.NoFocus)

main.py

from PyQt5 import QtCore, QtGui, QtWidgets

datas = {
    "Category 1": [
        ("New Game 2", "Playnite", "", "", "Never", "Not Played", ""),
        ("New Game 3", "Playnite", "", "", "Never", "Not Played", ""),
    ],
    "No Category": [
        ("New Game", "Playnite", "", "", "Never", "Not Plated", ""),
    ]
}

class GroupDelegate(QtWidgets.QStyledItemDelegate):
    def __init__(self, parent=None):
        super(GroupDelegate, self).__init__(parent)
        self._plus_icon = QtGui.QIcon('Ok.png')            # ('data/images/plus.png')
        self._minus_icon = QtGui.QIcon('ball.png')         # ('data/images/minus.png')

    def initStyleOption(self, option, index):
        super(GroupDelegate, self).initStyleOption(option, index)
        if not index.parent().isValid():
            is_open = bool(option.state & QtWidgets.QStyle.State_Open)
            option.features |= QtWidgets.QStyleOptionViewItem.HasDecoration
            option.icon = self._minus_icon if is_open else self._plus_icon

class GroupView(QtWidgets.QTreeView):
    def __init__(self, model, parent=None):
        super(GroupView, self).__init__(parent)
        self.setIndentation(0)
        self.setExpandsOnDoubleClick(False)
        self.clicked.connect(self.on_clicked)
        delegate = GroupDelegate(self)
        self.setItemDelegateForColumn(0, delegate)
        self.setModel(model)
        
# +++ vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv        
#-        self.header().setSectionResizeMode(QtWidgets.QHeaderView.Stretch)

        self.setColumnWidth(0, 20)
        
        self.header().setSectionResizeMode(1, QtWidgets.QHeaderView.Stretch) 
        self.header().setSectionResizeMode(2, QtWidgets.QHeaderView.Stretch)
        self.header().setSectionResizeMode(3, QtWidgets.QHeaderView.Stretch)
        self.header().setSectionResizeMode(4, QtWidgets.QHeaderView.Stretch)
        self.header().setSectionResizeMode(5, QtWidgets.QHeaderView.Stretch) 
        self.header().setSectionResizeMode(6, QtWidgets.QHeaderView.Stretch)
        self.header().setSectionResizeMode(7, QtWidgets.QHeaderView.Stretch)
        
        self.setFocusPolicy(QtCore.Qt.NoFocus)
# +++ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

        self.setSelectionBehavior(QtWidgets.QAbstractItemView.SelectRows)
        self.setStyleSheet("background-color: #0D1225;")

    @QtCore.pyqtSlot(QtCore.QModelIndex)
    def on_clicked(self, index):
        if not index.parent().isValid() and index.column() == 0:
            self.setExpanded(index, not self.isExpanded(index))


class GroupModel(QtGui.QStandardItemModel):
    def __init__(self, parent=None):
        super(GroupModel, self).__init__(parent)
        self.setColumnCount(8)
        self.setHorizontalHeaderLabels(["", "Name", "Library", "Release Date", "Genre(s)", "Last Played", "Time Played", ""])
        for i in range(self.columnCount()):
            it = self.horizontalHeaderItem(i)
#            it.setForeground(QtGui.QColor("#F2F2F2"))

    def add_group(self, group_name):
        item_root = QtGui.QStandardItem()
        item_root.setEditable(False)
        item = QtGui.QStandardItem(group_name)
        item.setEditable(False)
        ii = self.invisibleRootItem()
        i = ii.rowCount()
        for j, it in enumerate((item_root, item)):
            ii.setChild(i, j, it)
            ii.setEditable(False)
        for j in range(self.columnCount()):
            it = ii.child(i, j)
            if it is None:
                it = QtGui.QStandardItem()
                ii.setChild(i, j, it)
            it.setBackground(QtGui.QColor("#002842"))
            it.setForeground(QtGui.QColor("#F2F2F2"))
        return item_root

    def append_element_to_group(self, group_item, texts):

        j = group_item.rowCount()
        print(j)
        '''
        item_icon = QtGui.QStandardItem()
        item_icon.setEditable(False)
        item_icon.setIcon(QtGui.QIcon("game.png"))
        item_icon.setBackground(QtGui.QColor("#0D1225"))
        group_item.setChild(j, 0, item_icon)
        '''
        for i, text in enumerate(texts):
            item = QtGui.QStandardItem(text)
            item.setEditable(False)
            item.setBackground(QtGui.QColor("#0D1225"))
            item.setForeground(QtGui.QColor("#F2F2F2"))
            group_item.setChild(j, i+1, item)


class MainWindow(QtWidgets.QMainWindow):
    def __init__(self, parent=None):
        super(MainWindow, self).__init__(parent)

        model = GroupModel(self)
        tree_view = GroupView(model)
        self.setCentralWidget(tree_view)

        for group, childrens in datas.items():
            group_item = model.add_group(group)
            for children in childrens:
                model.append_element_to_group(group_item, children)


if __name__ == '__main__':
    import sys
    app = QtWidgets.QApplication(sys.argv)
    w = MainWindow()
    w.resize(720, 240)
    w.show()
    sys.exit(app.exec_())

После строки:

self.header().setSectionResizeMode(QtWidgets.QHeaderView.Stretch)

Добавьте вот эти две:

self.header().setSectionResizeMode(0, QtWidgets.QHeaderView.Fixed)
self.header().resizeSection(0, 25)