Does Store Operations include a feature to enable a Gift Registry by customer?

There is no gift registry features built in, but add-ins are available including one with a handheld scanner option.
You could alternately use Quotes to add gift items under a customer's name, then remove these items as they are purchased.
This works fine as a gift registry if you enter a reference or comment on the invoice (Shift-F9 Details) to identify it as a quote.
You can also set an expiration date on Quotes in Manager | Configuration | Ordering.

Note: While a workorder could also be deployed for this purpose, keep in mind a quote does not affect inventory levels while a workorder flags the items as committed.