When using the following method, the player defined in "player" will get the "testgroup" group but it will replace the group that the player was already in. Instead I want it to add the group as the method says.

perms.playerAddGroup(null, player, "testgroup");

I am using PermissionsEX as my permissions plugin. Please let me know how to solve this issue.

Please check with PermissionsEX on proper ways to add groups to players. iirc this may be a result of you trying to add a group to a player not bound to a world.. either way this is in the implementation of PermsEX not Vault.