Can a motorcycle license be obtained with a teaching permit?

Yes, it can.

A teaching permit can be granted at the earliest six (6) months before the student reaches the minimum age required for obtaining the respective driving license category (16 years for category A1; 18 years for category A2; 24 years for category A, or if the person has had an A2 category driving license for at least two years, 20 years).

Additional instructions: https://ajokortti-info.fi/fi/ajokortin-hankkiminen/opetuslupa