In Biblical times a father could actually agree to marry off his daughter under the age of twelve-and-a-half. (Deuteronomy 22:16). By the times of the Talmud the recommendation had become "she must be grown-up enough to agree to marry the fellow."
Yes, a previously-married woman is considered more "on her own" than the first time around. See for instance Numbers 30:10; a twelve-year-old girl was considered sufficiently tied to her father that he had the power to cancel out her vows -- but not if she was previously married.
(See Rabbi Aryeh Kaplan's footnotes in the linked texts for more sources.)
However this whole thing was very much tied to a time when most women were married right around their twelfth birthdays. Once a woman reaches twelve-and-a-half, she's completely independent. Her father can't marry her off, cancel her vows, sell her, or the like. (Though it's generally recommended for children of any age and gender to involve their parents somewhat in the process of finding a spouse.)
So there may be some cultural baggage behind the "father walking his daughter down the aisle", but you'd really have to squint and read things sideways to connect that directly to Jewish law. In most Jewish weddings today, the bride is escorted down the aisle by both her parents (just as the groom is escorted by his). Actually, in many ultra-Orthodox weddings, the bride is escorted by her mother and mother-in-law-elect, and the groom by his father and father-in-law-elect. It's all split by gender, none of this father-daughter business.