I don't think breastfeeding a 2 year old is strange at all!
Throughout history (and still in many parts of the world) breastfeeding was necessary until 3-5 years of age. This is because humans lived in periods of feast and famine. One day they'd have 100lbs of buffalo they needed to eat before it spoiled and then a few weeks later they'd be living off nothing but nuts and berries. That kind of up and down diet is very bad for a developing toddler. Kids need consistent fat and nutrients in order for their brains to develop properly. Breastfeeding made it possible to have a consistent food source available at all times. Even older toddlers who were capable of eating "real" food would still nurse so they could survive the periods of time that food was unavailable.
In developed countries like the US where we always have food just a grocery story trip away this isn't as necessary. Our kids have access to the food they need every day. Of course, this doesn't mean they CAN'T nurse past 2 years old. Toddlers are still built to nurse because that's how we evolved. It just depends on what you want to do. We're lucky that way to have a choice :)