She is still a child even though she has "Flowered" (Tryion says this when protesting the arrangement to his father), I would imagine being the same as an under-aged person in current times
Tyrion was referring to Sansa's childish naiveté. Tyrion is, by Westerosi cultural standards, an incredibly progressive character. Others do not share Tyrion's forward thinking.
Consider Danaerys' young age (in the books) at which she marries Khal Drogo; and no one protesting to this based on her age. While the Essosi and Westerosi may be different in regards to acceptable marrying age; I can't think of any case where age was an issue in Westeros. With the sole exception of Tyrion, who is unusually progressive for his time.
Being that her Mother is still alive at this time, it would only be logical that she is the only one who could decide who her daughter was to marry
Catelyn Stark specifically talked (I think to Sansa) about being married off to Ned, and not loving him in the beginning. Assuming Catelyn has always been truthful about that; this means that Sansa has grown up knowing that marrying someone and loving them are two separate things.
Furthermore, as a noble house's daughter, Sansa is well aware of the existence of arranged marriages. It's almost a given for people of her stature.
Being that she is captive, being made to get married under duress would not make the marriage official , which would mean Tyrion Lannister would have no claim to the North
She wasn't willing to marry Ramsay Bolton either, but that happened.
Keep in mind that if Sansa's surroundings say that the marriage was valid; then no one (without proof) can say otherwise. People from other cities or who were not present at the time have no way of knowing truth from a lie.
Just because Sansa did not like who she was expected to marry, does not mean that she doesn't begrudgingly agree to it. She is more than aware of the existence of arranged marriages as a noble house's daughter; and is also capable of understanding that her life could be made a living hell (even more so) if she refuses to comply.
Sansa's desire to go to King's Landing proves that she wants to be part of the royal court. By acquiescing to the Lannisters' plan for her to marry Tyrion, she can ensure that she stays there and is not sent away.
Edit
To summarize the comments below this answer:
You're ignoring the answers that everyone (both me and the commenters) have provided, which amply answers the question at hand. Either you are willfully ignoring the answers and this question has become pointless to answer, or you do not understand the answers and therefore do not understand the underlying premise of pretty much all events in Game of Thrones. Try to understand the difference between truth and the perception of truth, much of the Game of Thrones plot hinges on this.