Just because there are a lot of possible values doesn't matter. There are also a lot of possible pieces of data that could be fed into the hash function. In fact, an infinite number. Which is larger than 2^128. Unimaginably larger. Collisions will happen. Sure, intentional collisions might be next to impossible, but collisions will happen.

Consider a piece of data that is 128 bits long. There are 2^128 possible values. A 128 bit hash function will either yield a unique value for each possible data value (which will exhaust the entire range of unique hash values), or you'll have a collision. Now, lets add in pieces of data that are 129 bits (or any other length) as well - you'll have a collision immediately.

Perhaps I'm misunderstanding you, but doesn't the existence of collisions guarantee that h(1)== h(some_real_data) unless the hash function is hilariously broken?

Cryptographically secure hash functions state that it's unlikely to occur, but the fact that you can take data a greater size than the output will guarantee that you'll have collisions. If you have collisions and the hash function is generating output that is uniformly distributed across the possible output space, you will hit h(1)==h(some_real_data) at some point (edit: given that "some point" is a mathematical concept, not an actual point in time. Of course, there's no guarantee that we'll ever get collisions for hash functions with massive output spaces. But I'd argue that's a limit of computing, not of the math behind the functions themselves)