One station takes up one spot of your general item limit.
You need to be at the original station to bind a set to an attunable station. The binding doesn't work via the station menu but you have to stand in the area of the original station and activate it from your inventory.
Many thanks for this. If I have the tiny starter house, how many stations can I fit in it, assuming I don't have eso+?
Thanks.
It depends on which home you mean. If you mean the free apartment from the initial quest it will have space for 15 items. Other apartments have 50, small homes 100 slots.
Here is a link to a table which lists all homes and their item slots.