Certified GKH Gardening Expert
No, they are not. Balsam's botanical name is Abeis balsamea or Pinus balsamea, depending on who you ask. Lime is Tilia Europoea (though there are many species and varieties of lime, so this may vary.)
The first word in the botanical name will name the family. The second the species. So, as the botanical family names are different, they are not in the same family.