Olivia's status is conveyed in several ways: how she carries herself, how she dresses, her household, and how people treat her.
The simplest way is how people treat her. Duke Orsino is a nobleman; it is natural that his affections should fall on someone of the same class.
Olivia has the resources and leisure to go into mourning for seven years for her lost brother. To be blunt, you have to be really rich to do this. The poor have to keep working. Likewise, wearing that veil implies she has control over her self-presentation.
She assumes others will answer her commands, has servants at the ready, and enough money to keep a distant and separate household, one with enough economic margin to maintain Sir Toby as a freeloader. All of these show status.