NO! - they don't 'read', but they CAN recognise shapes, and see 'words' as shapes - but there is a very big difference between that and 'reading'.
Having worked in primary schools as helper and as TA for twenty-five years, I have been involved with maybe HUNDREDS of children learning to read. During that period of time teaching methods have changed greatly, and what was THE thing thirty years ago is now discounted as pretty useless!
Starting in the '60s, Dr Glenn Doman promoted his book 'Teaching Babies to Read', and his Institute is still operational today and claiming to 'teach' very young children things way beyond what one would normally expect. His method was basically 'flash cards'; we tried it in the '80s with our DS, when he was around two. He recognised a few words, but soon lost interest. Recognition of even long words with a distinctive 'shape' may be achieved, but there is unlikely to be any real UNDERSTANDING of meaning.
INVOLVING babies and children with books, stories, and 'mark making' materials can only be a good thing, provided they enjoy it, and no pressure is put on them to 'learn' in any formal way. When our DS was two, on a walk one day we saw a great green cricket on a hedge. I wasn't sure what it was, but said we would look it up in a book when we got home, which we did. Between two and five years of age the toddler brain is making 'connections' and the more that are made, via appropriate stimulation and experiences, the better, for future intelligence and learning ability.
Even today, using books for reference purposes is still a useful skill to learn, and one that is needed in school; the Internet hasn't totally replaced books [yet!]. The more relaxed a child is about handling and using books, when he starts school, the more progress he can make.