Yes, you can transfer from Anna University affiliated college to SRM University. But the process in itself is very hectic. You will have to go through it once or seek help from someone who has already transferred. 

You can transfer only during odd semesters. The university does not allow transfer during the even semesters. One of my sister's friends wanted to get transferred. So she applied for transfer during the 4th semester and got transferred in the 5th semester.