Is there a Boston equivalent to NYC's Sweetiepie?
Wow! That is a really beautiful space. I don't think you will find anything with quite that level of design concept in Boston, but one place that has a similar aesthetic on a (much) smaller scale is Sweet Cupcakes in Harvard Square (though they do not have a lot of tables). I don't know if the Back Bay location has more seats, because I haven't been, but I do like their cupcakes a lot and imagine a toddler would too.
There is also a place called the Wenham Tea House, which I have never been to but which does have some vaguely similar menu items (perhaps it was the Sweetiepie of 100 years ago?):
Lastly, the most ice creamy ice cream shop I have been to in this area is Christina's, but the ice cream far outweighs the atmosphere there.