The problems seems to be that it's not possible to emulate/modify the sector 0, which is often the UID (identifier).
This question is linked (but probably outdated).
NFC enabled phones can ONLY read NFC and passive high frequency RFID (HF-RFID). These must be read at an extremely close range, typically a few centimeters. For longer range or any other type of RFID/active RFID, you must use an external reader for handling them with mobile devices.
You can get some decent readers from a lot of manufacturers by simply searching on google. There are a lot of plug in ones for all device types.
I deal a lot with HID readers capable of close proximity scans of HID enabled ID cards as well as NFC from smart phones and smart cards. I use SerialIO badge readers that I load a decryption profile onto that allows our secure company cards to be read and utilized by an application I built. They are great for large scale reliable bluetooth scanning. Because they are bluetooth, they work for PC/Android/iOS/Linux. The only problem is, HID readers are very expensive and are meant for enterprise use. Ours cost about $400 each, but again, they read HID, SmartCards, NFC, and RFID.
If this is a personal project, I suggest just using the phone and purchasing some HF-RFID tags. The tag manufacturer should have an SDK for you to use to connect to and manage the tags. You can also just use androids NFC docs to get started https://developer.android.com/guide/topics/connectivity/nfc/. Most android phones from the last 8 years have NFC, only iPhone 6 and newer apple phones have NFC, but only iOS 11 and newer will work for what you want to do.
From a technical perspective there's nothing that would prevent you to implement a payment terminal using the built-in NFC reader capabilities. All Android NFC devices are capable of speaking the reader-side of the ISO/IEC 14443 contactless smartcard protocol (as used by EMV payment cards).
However, there are a few other things that may inhibt such efforts:
I personally think that the latter two were the main reasons for Square to provide their own device. Though this is just pure speculation and not quoting from any official sources.
Sorry to say this, but it seems to be a long way off before truly turning a NON NFC into NFC phone (as pointed out by Izzy in comments, there have been some efforts but no progress)
NFC SD and SIM Cards claims that Both SIM and SD cards can be equipped with NFC chips can be used to enable.
Toshiba has come up with NFC SD card last year, but the catch is that it requires NFC enabled device to work with. All it does , is enabling seeing the contents of SD card without physical connection
NFC-enabled SIM cards launched a couple of months back in Singapore for paying for rail travel- but they work only with NFC enabled phones