PCI DSS stands for Payment Cards Industry Data Security Standards. They are a global set of security standards adopted by the major card schemes (such as Visa and Mastercard) that you must comply with in order to process card payments. If you are PCI DSS compliant, it means you are implementing the necessary controls in order to maintain a secure environment for cardholders' data.
With respect to PCI DSS compliance, failing to migrate off a Magento 1 ecommerce website by June 2020 will cause merchants to fall out of PCI DSS compliance. This is because you will no longer be able to comply with Requirement 6, stating that you "develop and maintain secure systems and applications by installing applicable vendor-supplied security patches".
This information holds true irrespective of your acquirer or PSP.
Please refer to our Adyen Docs page on Magento 1 EOL for more information.