What does Portia think about all of this? Well, she's not happy:
[...] O me, the word 'choose!' I may
neither choose whom I would nor refuse whom I
dislike; so is the will of a living daughter curbed
by the will of a dead father. Is it not hard,
Nerissa, that I cannot choose one nor refuse none? (1.2.3)
For Portia, the lottery is a symbol of a dead father's control over his daughter's right to choose her own husband. Notice Portia's repetition of the word "choose" three times in this passage? Over the course of the entire scene, the word shows up no fewer than ten times, which emphasizes the point that Portia has no choice (read: power) here.
It turns out that parents (especially fathers) often got to decide who their daughters would marry in the 16th century, and we see a lot of this in Shakespeare. (In The Taming of the Shrew, for example, Baptista Minola arranges Kate's marriage to Petruchio without ever consulting his daughter.)