Is it permissible to shake hands with one's mother's paternal aunt, knowing that her mother and the grandfather's mother are different, and they are half-siblings from the father's side?

The relatives of the father or mother, however high up, are considered relatives of the children. So, the paternal aunt of the father, the paternal aunt of the mother, the maternal aunt of the father, and the maternal aunt of the mother are included among the unmarriageable relatives mentioned in the Holy Quran (An-Nisa: 23). It makes no difference whether they are full sisters, paternal half-sisters, or maternal half-sisters. The paternal aunt and anyone related through paternal aunts, such as the sisters of fathers and grandfathers, and the maternal aunt and anyone related through maternal aunts, such as the sisters of grandmothers – all of them are unmarriageable. Based on this, your paternal aunt mentioned is one of your unmarriageable relatives (mahram), and it is permissible to shake her hand.