Do storage encryption systems care about size of data? I was studying about Psuedorandom functions and their use as encryption functions. One of the things that I read was "birthday bound" or "birthday attack".  

When encryption is used for something like disk encryption, the amount of data being encrypted can be very large and over time be large enough to launch a birthday attack. Do storage encryption systems care about it? Or is it not big of a deal? Or is my assumption of data size being a problem, wrong?
It's common for encryption algorithm specs to specify a hard limit of how much data you should encrypt with the same key.  For example, consider the AES-XTS encryption algorithm, which was designed specifically for full-disk encryption:


The IEEE specification recommends that the length of a "data unit" (e.g., disk sector or block) encrypted with a single key should not exceed $2^{20}$ AES blocks.  
The NIST recommendation upgrades this to a hard requirement.


Since each AES block is 16 bytes, and $2^{20}$ bytes is 1 MiB, that's a limit of 16 MiB ($2^{24}$ bytes) per data unit.  You need $2^{44}$ such data units to hit the $2^{64}$-block birthday bound, but 1 TiB ($2^{40}$ bytes) is only $2^{40} \div 2^{24} = 2^{16}$ times the size of the 16 GiB maximum data unit.  So disks don't appear to be big enough yet to hit this limit.

But note that this limit is per encryption key.  If you use multiple encryption keys then you get to reset the limit for each key that you use.  It's already common to use individual keys for each disk volume, and some filesystems (e.g, Apple's APFS) already support per-file encryption keys.