My understanding is that there's a historical reason for this convention. Many telescope observations are calibrated by comparing the sky measurements against measurements of thermal loads with known temperatures. Since the calibration sources are specified with a temperature, it's straightforward to measure the source brightness in units of a pseudo-temperature. But I agree that it's not intuitive for those familiar with it.
Another layer on this is fact that doppler broadening of emission lines from molecules means they cover a finite width. Since typical velocities in astronomy range from fractions of a km/s to hundreds of kilometers per second, line widths are typically given in km/s. Since these line widths originate due to doppler motions, a width in km/s is equivalent to a width in Hz (for a given reference frequency for the spectral line).
The combination of these two things means people often report integrated flux measurements in units of K * km / s! It seems bizarre at first, but this is equivalent to specifying it as power / unit area (again, when there's a reference frequency specified). But these units are often convenient in their own way.