It is the job of both of you. However the motivations are different. Over time one or the other will be more useful to follow.

Your manager needs you to bridge the gap between when the company was doing yesterday (this is what is making the money today), and what the company will do tomorrow. When your manager sees that next year they will need a skill and gets you to develop that skill they have your domain knowledge of the existing product to apply to the next thing and the whole gets done faster because of your experience and new skills.

You are responsible for yourself. If the next job is a variation on what you did today no training is required so he won't give it - it makes you less valuable to outsiders so you won't leave. Also because you are not growing he doesn't have to give you top raises to keep you from leaving - which is more money to give someone else who is better than you. The downside of this is you need to guess where the next big thing will be and sometimes you will be wrong.