How do they know the newly-spotted gas near the center of our galaxy is molecular without knowing what gas it is?
They know there's molecular gas because they observed emission from the CO molecule in two of the previously identified (atomic-hydrogen-emitting) gas clouds. From the paper:

... we targeted two objects (hereafter, MW-C1 and MW-C2), highlighted
by red boxes in Fig. 1, in the 12CO(2 → 1) emission line at 230.538
GHz with the 12-m Atacama Pathfinder Experiment (APEX) telescope....
We mapped both clouds in 12CO(2 → 1) emission over a 15′ × 15′ field
centred on the peak of the H I emission,... These data revealed
molecular gas outflowing from the centre of our Galaxy.

CO emission is always assumed to come from molecular gas clouds which are, by mass, mostly H$_2$, though the conversion factor needed to turn a CO measurement into an H$_2$ measurement is a matter of debate (this is discussed in the paper).