Use 'tag onto' when referring to physically joining something to another thing.

Attach Physically

'Tag onto' can express someone joining a group in a social setting.

Social Inclusion

It's used informally to mean someone is invited casually or without planning.

It's meant to augment or give additional value to the existing item.

Can imply something was added later or as an afterthought.

The thing being tagged on is usually of secondary importance to the main item.