Does it make sense to say that characters in Pride and Prejudice are true to themselves?

Yes, it does. Elizabeth Bennet, for example, knows that she must marry for love, and she is unwilling to compromise her principles, even when presented with financially advantageous proposals from her cousin Mr. Collins and Mr. Darcy (the first time he proposes). Though either match would make her mother so happy and would, at least, secure her family against future financial strain, she cannot bring herself to entertain either of these men when she does not love them.
Charlotte Collins, née Lucas, on the other hand, really doesn't care about love. As she is nearing spinsterhood, financial security and avoiding becoming a drain on her family are her priorities. Therefore, when presented with a marriage proposal from Mr. Collins—a man she knows cannot possibly love her, as he just proposed to her best friend a few days earlier—she accepts with alacrity. She has said more than once that she is "not romantic," and she proves it by marrying someone she does not love so that she can maintain respectability and peace in her own life. These are her priorities, and she acts in accordance with them.
We might also consider Lady Catherine. We may not like her very much, but we have to concede that she sticks to her principles as well. Family is her priority, and when she hears a rumor that her nephew—who she intends her own daughter to marry—has proposed to a relatively penniless young woman with questionable relatives, she flies into action. In her own way, she is attempting to protect her family, and she shows her continued displeasure by refusing to attend the wedding of Darcy and Elizabeth. Again, we may not like her, but she sticks to her guns, so to speak.
Thus, these characters, and others, do behave in ways that feel "true to themselves." Most of them are fairly uncompromising of their principles, and when they do seem to change—as Elizabeth and Darcy do—it is due to a sincere reassessment of their prior behaviors and not a failure to be true to themselves.