Sure, they can measure different coordinate times. Those times can be a component of the invariant spacetime interval, without being invariant themselves. Different observers have different components that combine to the same interval.
It's the proper time that is the invariant length of a spacetime interval. Everyone agrees on the time that a clock measures on its world line between two events. But to different observers, the clock's path with have different coordinate times, and different coordinate lengths. Consider an infinitesimal line element of such a world line. For some observers, the clock can be stationary over that line element, with spacial components of 0 and an infinitesimal time component. For other observers the clock is moving, and the line element contains infinitesimal spacial component. They disagree on the components, but agree on its invariant length that results each set of those components.