Display node local tasks on custom page
Ok actually this was a stupid question...
I was trying to add the node tabs on my custom page without adding a tab to this custom page on the node view itself (entity.node.canonical)
Simply adding the new local task on the node view automatically displays other tabs when viewing any of the tabs
my_module.links.task.yml :
my_module.my_custom_tab:
  route_name: my_module.custom_page
  title: 'Custom'
  base_route: entity.node.canonical