This is generally correct, however, exception can be applied in cases of representatives, which act only on behalf of one of the parties (payer or the payee), in case there is no need for license in certain jurisdictions.If we're talking about EU/EEA, any person (legal entity or natural person) who holds, transmits, or processes money on behalf of an unrelated person is offering a financial service and needs a license. Of course, there are payment agents and billing companies that violate this rule and face no repercussions.
In some cases, billing companies are subsidiaries (or agents) of financial institutions and are set up to keep the activities at slight distance from the financial institution itself. Quite common in adult entertainment.
This is because although payment services are regulated by Payment Services Directive, the EU states have interpreted and transposed it slightly differently in their legislation.