All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
If you are sure that this grave is your mother's grave, then you can contact the family of the other woman to clarify the mistake to them so that they would move the tiles to their mother’s grave. However, if it is left as it is, then there is no harm in that if the tiles do not cover a large part of the grave, because building a wall or fence around the grave with stones for a need is permissible and it is not considered as building on the grave.
The fatwa of Shaykh Ibn Baaz reads, “The graves that are abused by people or animals walking on them, or garbage is thrown on them, or people relieve themselves on them, then they should be surrounded with a fence or a wall; the people of the town should cooperate in building a wall around them that prevents animals and people from walking on them.”
You should consider the extent of the damage that is expected to be caused to the grave if the tiles are removed; if the damage is considerable such that the body of the deceased would be exposed or harmed, then you may keep the tiles in place and compensate the people of the other grave for them, because the sacredness of the Muslim is confirmed both while he is alive and after his death.
‘Aa’ishah narrated that the Prophet said, “Breaking the bones of the dead is like breaking the bones of the living.” [Abu Daawood and Ibn Maajah]
Ibn Hajar said, “This hadeeth indicates that the sacredness of the believer after death is still confirmed as it was during his lifetime.”
However, if removing the tiles would only cause removal of some mud or simple scratching, then there is no harm in removing the tiles if their owners wanted to remove them, and if they agreed that you give them compensation and keep the tiles, then this is better if you can do so.
It should be noted that there is nothing wrong with writing (on the grave) that does not include any praise or exaggeration of the deceased.
In Shaykh Ibn 'Uthaymeen's commentary on Riyaadh As-Saaliheen, he said:
“There are details regarding the ruling on writing on the grave: as for the writing that is only intended to confirm the name of the person in order to know his grave, then there is nothing wrong with it. As regards the writing that resembles what the people in Jaahiliyyah (Pre-Islamic Era) used to do, such as writing the name of the deceased along with praises of him, that he did such-and-such and other praises, or writing poetry about him, then this is prohibited. Included in this is what some ignorant people do, that they write Soorah Al-Faatihah (the opening chapter of the Quran), for example, or other verses of the Quran on the gravestone; all this is forbidden, and whoever sees this on the grave must remove this stone, because this is among the evil that must be changed.”
Allah knows best.