Age? There are special growth related issues for adolescents but I don't know their symptoms on understand how they are associated with tennis. Growth plate
injuries affect junior baseball pitchers before their growth plates have finally solidified. When the growth plates solidify into bone it largely ends bone length growth.
See links related to elbow injuries and also any related to elbows and special growth injury issues for adolescents.
I once had an adult friend who had persistent elbow pain. We took high speed imaging of our serves. You could easily see why he had pain. His elbow was slightly over extended - past straight when he impacted the ball. And the ball impact force was in the direction to directly add to the over extension. He had developed his own serving technique and it stressed his elbow by hyper-extending it.
Get a video of your serve, preferably a high speed video, and compare your technique to proper serves.
Unless your coach is an orthopedic Dr don't accept his medical evaluation of your elbow. Injury diagnosis requires a Dr.
Discussion of elbow injuries. The table injuries next to "throwing" might apply to serving as well.
See a Dr. Please don't play with pain.