If both husband and wife enjoy sex. Then why does husband has to give mahr to get right to have intercourse, if it is a mutual right?

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.

The dowry is an obligatory right upon the husband to his wife, and this dowry is enjoined by the Sharee’ah.

Allah Says (what means): {And give the women [upon marriage] their [bridal] gifts graciously.} [Quran 4:4]

Al-Qurtubi quoted in his Tafseer (interpretation of the Quran) from Qataadah that he said: "Graciously means an obligatory obligation."

So it is her right from the start.

It is not just a compensation for enjoyment, but there are many Islamic objectives related to it, which were mentioned by the scholars, including:

1) Honoring the wife and showing her honor and status.

2) The person giving this money (dowry) shows his desire for marriage.

3) The durability of the marriage: if the dowry is not obligatory, the husband would not care about separating his wife for any futile reason.

For more benefit, please refer to Fataawa 245665 and 59310.

Allah knows best.