and in 'nowdayish' it's a store

Because "that" is a demonstrative pronoun.

Turning back to ez and az, i.e. the demonstrative pronouns, there are two things to keep in mind. By Hungarian logic they make their object definite, therefore you need to add a definite article right after them. Examples:

ez a ház - this house

az az asztal - that table

ez a férfi - this man

az a nő - that woman

ez a Péter - this Péter (e.g. if there are more Péters in a group we’re talking about), or ‘this Péter guy’

az a szép lány - that pretty girl

ez a gyors kocsi - this fast car

The second important rule is that they are in the same case and number as the nominal they are referring to. The previous list in plural:

ezek a házak, azok az asztalok, ezek a férfiak, azok a nők, ezek a Péterek, azok a szép lányok, ezek a gyors kocsik