Which of these two methods is standard when creating a 1 to many database relationship?
Alternatively I can create multiple addresses and give them all the same AddressId.


This is not a good idea at all. The AddressId supposes to be a primary key in your Address table.

Foreign Key

Use foreign Key relationship between customer and address tables, if customer may have multiple addresses associated to him/her. 

Do enforce foreign key relationship between customer and its addresses by CustomerId being a primary key in Customer table and foreign key in Address table. This will ensure that no orphan records will be left, when customer record is deleted or in-activated.

One-to-many relationship diagram:



In the case if same address can be assigned to different customers, aka many-to-many between customer and addresses, do use a third table CustomerAddress (junction table). Junction table should save pair of (CusomerId, AddressId) and any other related information specific for this combination.

For example, in below diagram we have many to many relationship between Orders and products.

For more references on DB design patterns look here.

Many-to-many relationship diagram: