Yes, a student can be in more than one class at a time. Each teacher will be able to see the student in their class and on their reports if a student is in multiple classes. Any teacher can change the student's journey.
If you're in a school that is rostered, the data table that is sent to ST Math has to list additional teachers as co-teachers. Check with your school office for more information.
If your students use picture passwords, you can add any student to your class either by using an Invitation Code or by searching for the student. The instructions are the same as for transferring students and can be found in this article: How do I transfer student to a different class?