Yes from my point of view and eating both Linguine and Spaghetti the linguine does indeed taste different than spaghetti.
It's not because of the shape of the noodles though because shape of noodles cannot alter the taste of the linguine or spaghetti.
But I believe the reason that different noodles taste different from each other is in the way they are cooked and the length of time they are cooked.
Some noodles may require cooking longer than others which can likely affect the taste of the noodles some.
You shouldn't notice too much of a difference in the taste of the noodles between linguine and spaghetti but there can be a different taste.
Some people say they taste the same but everyone has different taste buds which can make something taste different.
The sauces you use can also affect the taste of the noodles you cook and eat.