I subscribed for Apple Music and I got just one month free trial instead of three

Where are you (region/country)? As the offer is not the same worldwide.

After signup, you would have received a confirmation email with the subscription details, including duration of the trial period.

On the subscription page for your account, the subscription might be listed as “Individual (1 Month)”. That is not about the trial duration, but the subscription period/commitment after the trial, as opposed to a year contract.