Fanaelialae
Legend
Yeah, your player is in a bind.
That UA Ranger is only designed to go to level 5 AND isn't designed to be multiclassed. That means, unless you give him special dispensation, he is stuck at level 5 forever.
Remind him of that and he'll likely be willing to switch over to the updated UA ranger.
It isn't designed to be multiclassed into, but you can multiclass out of any class (assuming the DM permits it). There's no RAW reason he couldn't take 5 levels of this variant and then take his 1st level of druid (or whatever) at character level 6.
You just can't multiclass out of a class and multiclass into a variant of that same class (because it doesn't work - if you start from level 1 you are weak because you have a lot of duplicate features, and if you start from level 6 you have nothing to gain when you would acquire new subclass features and might lack some low level features that later features are based on).