It is possible to move faces together, as solid island, but not as separated:
In sync mode.
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There are a couple of ways to select entire islands for manipulation.
First, we could just select one face, then select linked, in the UV editor. This will select all faces that are part of the same island. Select linked, for me, is ctrl L. (We could also sync our UV selection and select linked in the 3D editor, limiting by seams in the operator panel, if our seams represent our current islands.)
Second, we could island select mode. Currently, we can see that you're in face select mode, with synced selection enabled. If you disable synced selection, there will be an extra selection mode enabled, which is island selection mode. In this mode, you can select and manipulate entire islands:
It it moves like one the left, but you want it to move like on the right, change Sticky Selection Mode in top-left corner:
If you just want to select more faces more easily, as Nathan says, hover over an island and press L or change selection mode to islands, or after selecting a single face, press Ctrl+ Numpad Add. If that still doesn't work, you may want to merge vertices (in edit mode in 3D viewport editor, not UV editor), by pressing A, M, B and maybe also F9 and adjusting distance.
Markus von Broady has the true-to-Blender answer. However, IMO, Textools is a must-have addon for UV unwrapping in Blender. Almost like Node Wrangler, like if you don't have that enabled then what are you even doing basically.
This addon comes with things like "Select By Island" as the default UV select mode.
Not to mention two dozen other awesome features built-in.