She's not necessarily too young. It depends on the child; two of mine were early readers at 3, and fully fluent by 4 or 5. The trick is to let her push things forward with her interest, her questions and her observations, and to not hold any expectations about her rate of progress or the route she'll take to get there. So I don't think you want to jump in and administer some sort of systematic learn-to-read program. It's possible she won't be ready to read for another three or four years, or that the best route to reading for her brain just now will turn out to be a route you hadn't imagined. (My own eldest dd learned to read by poring over mail-order catalogs and National Geographics silently for hours and weeks on end, while I had assumed she'd by nature be a systematic code-breaker wanting more of a phonics approach.)
It sounds like right now your dd is using her aural memory to full advantage, repeating things she hears you read. It's hard to know from your description whether she's connecting those words meaningfully with their printed representation on paper. But she may be, or she may begin to. A lot of children learn to read by memorizing simple (often repetitive) books and then running through the text repeatedly by memory while gradually noticing what the associated printed words look like.
At any rate if she's enjoying what you're doing now there's nothing wrong with it and it may very well help her pick up the building blocks of reading. You might try using some simple repetitive books to limit vocabulary and allow her to begin to see patterns in the text she's gradually memorizing. Stuff like Dr. Seuss was enjoyed by my kids. She might also enjoy playing 'word hunt' after you've read a short section of text. See if she can find the word "fish" (or whatever) where it's written on the page. Give her clues if she needs them.