That's what it means in this context. Usually "și" corresponds to the conjunction "and", and in this context it sort of carries the same meaning. For example:
They have strawberries and bananas. We also have dessert with chocolate.
Ei au căpșuni și banane. Noi avem și desert cu ciocolată.
Like "also", "și" marks that we have all the things they have plus dessert with chocolate. You can also do this by repeating the list and adding emphasis on the conjunction:
They have strawberries and bananas. We have strawberries, bananas and dessert with chocolate.
Ei au căpșuni și banane. Noi avem căpșuni, banane și desert cu ciocolată.
I hope this makes it less counter-intuitive.
That word order seems somewhat awkward in English. But I disagree with potestasity that "also" in English necessarily implies an additional dessert. It can also be understood as implying "we as well as they have a dessert with chocolate."
And I think what you are trying to do is differentiate between also applied to "we" and also applied to "dessert".
I wonder if the Romanian și in this context can be also be understood to apply either to "Noi" or to "desert"? Or would Romanian place the "și" in front of "Noi" if it was intended to refer to "we as well"?