Is it permissible for him to give the zakah of his wealth to his fiancée whose family are poor?

Firstly:

The scholars (may Allah have mercy on them) are agreed that it is not permissible for a man to give obligatory zakah to one on whom he is obliged to spend. For more information, please see the answer to question no. 81122 .

The fiancée and her family are not among those on whom the suitor is obliged to spend, so if they are poor and in need, it is permissible for him to give zakah to them.

It says in Sharh az-Zarkashi ‘ala al-‘Khuraqi (2/429):

‘Abdullah ibn ‘Abbas (may Allah be pleased with him) said: If they are relatives who are not your dependents, then give them the zakah of your wealth; if they are your dependents, then do not give them [zakah], and do not give it to those who are your dependents. End quote.

If you give zakah to her, then it becomes her property and she may spend it on what she needs, whether that is the costs of completing her studies or of preparing herself for marriage.

But that is on condition that she does not buy things that the husband is obliged to buy, so that the husband will not be giving zakah instead of buying some of the things that he is obliged to buy.

Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:

If a person receives money in a legitimate manner, then it becomes his property and he may dispose of it however he wishes in ways that Allah, may He be glorified and exalted, has permitted."(Fatawa Nur ‘ala ad-Darb).

We have noted previously, in the answer to question no. 21975 , that it is permissible to use zakah funds to help someone to get married, so long as that is on a reasonable basis, without any extravagance.

And Allah knows best.