Yes, I find this frustrating too. There is a code way to try and fix it which can be run at the python window or added as a macro - see instructions at this answer [https://gis.stackexchange.com/questions/183808/changing-default-folder-for-saving-layers-in-qgis?rq=1][1]:

from qgis.core import QgsProject
from PyQt5.QtCore import QSettings
prjhomepath = QgsProject.instance().homePath()
if os.path.exists(prjhomepath):
    QSettings().setValue('/UI/lastFileNameWidgetDir',prjhomepath)
    QSettings().setValue('/UI/lastRasterFileDir',prjhomepath)
    QSettings().setValue('/UI/lastVectorFileFilterDir',prjhomepath)
    QSettings().setValue('/Processing/LastInputPath',prjhomepath)
    QSettings().setValue('/Processing/LastOutputPath',prjhomepath)```

The code changes various folder locations stored in your QGIS.ini file to your projects home folder. There may be other folders to change - take a look at QGIS3.ini in a txt editor

  [1]: https://gis.stackexchange.com/questions/183808/changing-default-folder-for-saving-layers-in-qgis?rq=1

Yes, I find this frustrating too. There is a code way to try and fix it which can be run at the python window or added as a macro - see instructions at this answer:

import os
from qgis.core import QgsProject
from PyQt5.QtCore import QSettings
prjhomepath = QgsProject.instance().homePath()
if os.path.exists(prjhomepath):
    QSettings().setValue('/UI/lastFileNameWidgetDir',prjhomepath)
    QSettings().setValue('/UI/lastRasterFileDir',prjhomepath)
    QSettings().setValue('/UI/lastVectorFileFilterDir',prjhomepath)
    QSettings().setValue('/Processing/LastInputPath',prjhomepath)
    QSettings().setValue('/Processing/LastOutputPath',prjhomepath)```

The code changes various folder locations stored in your QGIS.ini file to your projects home folder. There may be other folders to change - take a look at QGIS3.ini in a txt editor.