Looks like a papaya ....

there is a houseplant forum .. for further learning ... if it had been taken outdoors in my MI ... 3 weeks ago ... it would have a lot more problems than yours .. cold damage ... also.. its stressed from the change of circumstances.. e.g. going from the growers greenhouse ... to the seller.. and to your house ... so a little leaf loss is not that big a deal ... also .. in fall.. many plants lose their older leaves ... due to the changing season ... anyway ... its stressed.. and not looking all that bad ... just water it properly .. and it should put out some new leaves ... and being stressed.. does not mean its hungry .. fert is not a response at this time ... if you just got it from the store.. i am sure there is enough fert in that pot for months ... in spring .. when the growing season starts.. then some fert might be helpful ... ken ps: its not suffering.. its just reacting to change .... you are suffering.. try not to love it to death ...

It looks reasonably Magnolia-like to me as well, but I couldn’t ID the cultivar from this distance and without flowers. Magnolias vary a fair amount in hardiness depending on cultivar. Regardless, if you are much colder than zone 6, growing in containers doesn’t work well for most plants because the roots get much colder than in the ground when the containers are so exposed to cold air on all sides, unlike roots in the ground that have more natural insulation. where are you generally and what is your USDA growing zone? Are you planning to grow plants in a container or in the ground? The NYC area is zone 7, and potted plants there should be at least hardy down to zone 5 if they are likely to survive. So it might be one of the smaller, hardier magnolias like ‘Jane’.

Possibly some type of Curcuma ginger. There are many hybrids. Check out Gingerwood Nursery online and look at their Curcuma pics and see if you can find a match.