The origins are different but both automate card data entry and establish “card present” pricing with the card schemes which is priced lower due primarily to lower fraud risk.
iZettle provides for chip and PIN user authentication where most debit cards in Europe were enabled for PIN only, without the possibility for manual entry. Thus, it enables merchants to accept EMV cards using devices that did not have secure chip readers and pin entry devices.
Square first solved for enabling merchants to “acquire” magstripe cards qualifying for card-present risk scoring. Users most often signed the screen to authorize the charge. Square now has an EMV reader and PIN entry device as well.
At least for Square, the transactions are routed via the hardware provider gateways making them the acquirer of the merchants’ transactions.
There are very strenuous requirements for PIN entry and EMV card/pin authentication. Ordinary phones and tablets are not designed to these standards.
NFC does open the possibility for low-value payments. Apple does not enable/allow payments using NFC esp with third party apps. Other hardware makers allow NFC based payments, but there are very strict rules around creating and delivering card credentials to a phone for payments. The major card schemes are providing this tokenized card data a service to card issuers.