Wow. This is a huge question. I will hit this from a few different sides, as I don't think a real answer can be given in a format shorter than a research paper.
As a semi-professional wine taster, I can tell you that in this world tastes are often grouped along lines that, while not matching botanical classifications, resonate with it. Every professional wine taster will tell you that the following represent flavors shared by all members in the groups: Berries, dark berries, red berries, red fruits, dark fruits, citrus, tree fruits, tropical fruits, stone fruits, mushrooms. The overlap in these is informative. Citrus fruit is a family of flavors, with a profile. Tree fruits (while all citrus fruits fit in this category) are different. These are the flavors that oranges and lemons, etc, share with apples and pears and other fruits that grow on trees. Citrus and tree fruit are very distinct profiles. A plum is a dark fruit, a tree fruit, and a stone from (because of the stonelike pit), but tree fruits, stone fruits and dark fruits all taste different from each other. Anything decribed as tasting of those categories tastes like a plum, though they don't taste like each other.
Next, the questioner and myself have be using sloppy nomenclature. An AROMA is what you can smell, with your nose. A TASTE is what you can taste, with your taste buds. What we are talking about is FLAVOR, a fusion of the two.
This matters, because food science now knows there are 5 known tastes (that your tongue can sense) : sweet, sour, salty, bitter and umami (sic). The last is only recently accepted as scientific fact, and it represents a taste best described as 'savory' or 'meaty'. If you want to taste it, find a recipe for 'dashi', or find some dashi, and taste it. Umami is the taste imparted by MSG.
That being said, potatoes and tomatoes do have the same primary taste: umami. This is not my opinion. Many scholarly writers on the subject of umami use potatoes and tomatoes as their examples (feel free to add references, I can't right now.)
So, I would say the answer is yes and no. All members of a family will share certain chemical traits, which will likely affect your tongue and nose in similar fashion, but these similarities are not a primary reason the families were divided as such. Some shared family traits likely are not discernable by your senses. Lots of food from many different families taste like chicken.
My final answer: mainly yes. Foods from the same families will likely have many of the same olfactory-active and taste-active compounds, though those grouping plants into families did not have these concepts in mind when creating the categories. Related plant TEND to smell/taste alike, as (mammal) family members TEND to look alike. Put in genetic terms, similar genotypes tend to express as similar phenotypes. But it is not a hard and fast rule.
As far a fruit vs vegetables: remember that the fruit and the vegetable just represent different parts of the plant. The potato and tomato are both night shades, but the tomato, being a seedpod, is a fruit, while the potato, as a root, is a vegetable. But I think the same rules apply. Rhubarb and sorrel are related, and have similar taste AND flavor. Broccoli, cauliflower and cabbage are all related, and they definitely smell/taste alike. So, here are some questions I don't know the answers to that may shed light. Are fennel and anise related? Tarragon and parsnip? Cilantro and whatever we put in soap?