Hope this simple demonstration makes you clearer. The index of table X will have both values, and the index in table Y will have only the value is_unique.
create table X (
id int CONSTRAINT x_is_unique UNIQUE
)
create table Y (
id int
)
create unique index y_is_unique on Y(id)
select name, is_unique, is_unique_constraint
from sys.indexes
where object_id in (object_id('X'), object_id('Y'))
and name is not null
drop table X
drop table Y
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