iOS itself does not provide a general facility for an iPhone to send a record of how many screenshots you have taken to a third party or to another person. Therefore, if someone claims that they can see the number of screenshots taken on your iPhone, it is important to first determine exactly how that information is supposedly being obtained.
If Snapchat itself is reporting your screenshot activity to another person, that would be a function of the Snapchat app/service rather than a normal iOS feature. It would be worth checking Snapchat's settings, account security, and contacting the app developer to understand what information is being shared and under what circumstances.
Another possibility is that the iPhone is managed by an organisation, such as a school, college, or employer. Apple supports Mobile Device Management (MDM), which allows an organisation to impose certain restrictions and configurations on an iPhone. For example, an administrator can configure a managed device so that screenshots and screen recordings cannot be taken. However, this does not by itself mean that the administrator receives a count of all screenshots taken.
You can check whether your iPhone has an organisation's management profile by going to:
Settings → General → VPN & Device Management
If a management profile is installed, open it and check which organisation is managing the device and what restrictions or configurations have been applied. Apple specifically recommends checking this section to identify installed management profiles.
If this is a company- or school-owned/managed iPhone, contact the organisation's IT or system administrator before making any changes. The administrator can confirm what is being managed and whether any monitoring or restrictions have been configured.
If this is your personally owned iPhone and you have no reason to expect it to be enrolled in an organisation's device-management system, first check Settings → General → VPN & Device Management for an unfamiliar profile. Also review your Apple Account security and installed apps.
If you find an unknown management profile or have strong reason to believe the device has been compromised, a full factory reset (Erase All Content and Settings) is a reasonable security measure after ensuring that you have a current backup and that you can recover your important data. If the device is genuinely organisation-managed, however, do not erase it without first speaking to the administrator, because the device may automatically be re-enrolled or you may lose required organisational access.
In short, an MDM profile is a possibility worth checking, but the mere fact that someone knows about screenshot activity is not, by itself, proof that the iPhone is jailbroken or MDM-managed. The source of that information needs to be established first.