Because voting is the most powerful action you can take (legally) in a democracy and the responsibility on both the voter and the government to make that a transparent, effective, effortless, verifiable and trustworthy transaction is tremendous. Working in banking, the regulations around identity are very strong - because the common person needs to know that their money is protected otherwise the banking industry would collapse. A vote should be treated with a similar respect. In economic terms, we've devalued a vote to nothing; there's no 'cost' to it, so people assign consider it valueless. There's no reason to protect it, no reason to fight for it and as such can be (and has been) taken away.
Ideally every person of voting age and ability would be provided at no charge an identity card attached to no database, no tracking that could act in replacement to commonly accepted identification methods such as driver's license, birth certificate, etc. You go through more identification to use a credit card at Walmart than you do to vote.