In terms of their goals yes. In terms of their impacts, it is an enormous difference.
However if one wants to be stupid in this country and form a union under a private employer there’s no reason they should not be able to.
They only hurt themselves and one company in the private sector. All the rest of their fellow citizens are not on the hook paying for them. They only have themselves to blame when the company lays them off or goes bankrupt and the union contract dissolves.
Public unions screw all of us. Job security of public employees is far stronger than private.
There are huge, real differences.
A government union is a double whammy, that's true. But that doesn't make private unions any better. They are vehicles for graft and politics, at the expense of the legimiately motivated and talented.
I hate ALL unions. Always have, always will.