What should I do if my current manager seems to have everyone on a regular schedule and I’m the only one “on call”?

You probably can't do anything. Subject only to an employment contract (if any) and the laws against employment discrimination (see below), employers are free to define their employees' responsibilities, to set schedules and duties, and to treat some employees differently--even worse--than others.

An employer cannot discriminate against an employee due to his or her race, religion, age over 40, disability, or sex--under federal law; your state may add a few other protected categories, such as sexual orientation. That does not mean, however, that, for example, a disabled, 50-year-old, African American Muslim female employee could not get different or worse treatment than other employees--just that she couldn't get different or worse treatment due to her age, disability, race, etc. But if there is a non-discriminatory reason to treat her differently--such as a lack of seniority or time in service--that is valid.

In your case, if you have only been there one month, then even if you did belong to a protected category, the fact that  you have little seniority could be a valid reason for making you be the on-call person.