What is an inlay apartment?

I'm not sure what an inlay apartment is but I think you live in a paired-home or a duplex apartment.

A paired home is two homes that share a wall and have opposite side entry.

or

duplex apartment / a two-storey flat

Two-storey flat

an existing small house or cottage that has been lifted up to accommodate the creation of a new basement floor housing a separate apartment, then set down again; thus becoming a modest two-story flat.

It only just occurred to me now, weeks later, that inlay apartment is almost certainly a typographical error for in-law apartment. The answer can stand.

I did a web search on floor plans for Einliegerwohnung, and based on that I think I can provide an answer.

The above seems to be typical; in other plans, the main unit and the secondary unit may share an entrance vestibule or foyer, but generally the living spaces are self-contained, and they do not fall under other categories:

• They do not share living spaces like a kitchen or toilet, and cannot be accessed internally from one another, so it is an apartment or flat as opposed to a guest suite.
• They are not equivalent, so they are not simply two halves of a duplex or paired home.
• The secondary unit is purpose-built as such, so it is not simply a large house that has been subdivided into separately rented areas as a multiple-unit dwelling.
• They are attached, so the smaller unit is not a guest house (in the sense of A small, separate house on the grounds of a larger one, used for accommodating guests, ODO) or laneway house.

The smaller or secondary unit may be known as a granny flat, an in-law apartment, or an accessory apartment among many, many other terms which vary slightly in usage depending on the locale. Such terms reflect a typical kind of resident— one who does not want or cannot afford a main residence, but can make use of a smaller one.

A common industry term appears to be accessory dwelling unit (ADU). Wikipedia uses the term secondary suite, though in the U.S. there is a regulatory distinction between an accessory unit and a secondary unit.

But in my experience with both landlords and tenants of such units, I would say there is no special term in common usage. No one advertises an "accessory dwelling unit for rent" on Craigslist, or says they're looking to lease an ADU for graduate school. The space is simply a flat or an apartment or a suite. It may be described by its location— basement flat, carriage house apartment, etc., or as an apartment attached to someone's house— but it will otherwise be understood from context whether the apartment being rented out is on the same property as the owner's living space or not.