Yes, it appears to be southern bush honeysuckle, probably cool splash variety. The following website gives further information. It would be okay to move it, although it usually will flourish in a variety of soils and moisture levels. Just be sure to prepare the site ahead of time and move as much root structure as you can. Water well to minimize transplant shock. See: https://extension.umn.edu/trees-and-shrubs/bush-honeysuckle