If you are in the EU, or your players are in the EU and your service is targeted at some part of (or all of) the EU, then this pretty clearly falls within the scope of the GDPR. Such a service would be collecting data associated with natural persons. At least some of it would be made public along with an online identifier, and other parts would be processed and stored, even if disclosed only in anonymized form. (By the way it is a non-trivial task to anonymize data well enough that it is no longer persona data under the GDPR. But let's suppose yoru methods are good enough.)
Under GDPR Article 6 the DC must have a lawful basis for such processing. If the basis is "consent" then the consent must be freely given, whch means giving consent may not be a condition of using the service.
Under GDPR Article 13 a notice must be sent to the Data Subject (DS) when the data is collected from the subject, including some 11 items or categories of information about the collection process, the Data Controller (DC), the purposes of processing, the legal basis of processing (which the DC must define), who will receive the data, the data retention period, and various rights of the DS.
Such a service must be prepared to respond to data access and data erasure requests, under articles 15-17, and data restriction requests under article 18. It must provide a method for a DS to easily make such requests.
Other GDPR provisions may well apply in addition to these, but that is enough to show that such service is very likely to be within the scope of the GDPR, unless none of the DSs are present in the EU or the related market area.