Q: Is the Trinity of Christianity and Hinduism the same? Or is the Trinity of Christianity and the Trimurti of Hinduism just a conicidence?
A: You are right to observe that there is an apparent similarity in the number three being related to both Christianity in the Trinity and to Hinduism in the Trimurti. Your question is a fine one and one we often have asked of us. The nature of the two similarities in this case is one that disappears as soon as we begin to look deeper.
The basic contrast is that with the Trinity, the concept is of one God, exiting in three dimensions or personhoods. The point is that the Trinity is not a concept of three gods, but of one God. In Hinduism we have three gods who serve as representation of either three gods or one God (Brahman), depending on which school and tradition of Hinduism you would choose to examine. Essentially, the Trimurti is one of the scales of Hinduism one might stop at to descriptively explain the Hindu gods. One could stop at one God (Brahman), three gods (Trimurti), move out to the 300 most popular gods, or move even further out to 3,000 --and so continuing on. With the Trinity there is not optional way to describe God. He is one God in three person, period.
The oneness of the Trinity is made more clear when we find this oneness is represented in three persons (Father, Son, and Holy Spirit), and that these three persons do not change over time. Neither is there the possibility of this changing to two, four, or seven persons in this representation. The flexible concept of the Trimurti is more a descriptive representation of the present status of popular gods and the fashioning of these three gods into a sort of three-some, where one god (Brahma) is the creator god, one god (Vishnu) is the sustainer god, and one god (Shiva) is the destroyer god. Originally, the most popular gods (of the Vedas) were Indra and Agni, with numerous deities vying for a popular third position (Varuna, Soma, etc.). There certainly was not the present concept of a Trimurti present until the later development and popularity of Vishnu and Shiva. There is essentially no worship or cult of Brahma (the first god of the Trimurti), and many Hindus do not worship either Vishnu or Shiva. This is all to say that the concept of the Trimurti is a non-essential and descriptive observation about Hinduism, not a doctrine of the very nature of God Himself. In contrast to this, the Trinity is a very essential teaching of the very nature of God Himself.

Here is a summary of this comparison:
TRIMURTI TRINITY
# of gods 3 1
# of persons 3 3
Names of Gods change? Yes No
# of Gods possibly different?
Yes No
Essential in worship? No Yes