Are "if" and "iff" interchangeable in definitions?

As it is a definition, the validity of the property (here "being a basis for the topology") must be defined by it in all cases. So implicitly all cases not mentioned do not have the property.

This convention is even stronger than the "if" meaning "iff" in definitions. Take as example the definition "an integer $p>1$ is called a prime number if it cannot be written as a product $p=ab$ of integers $a,b>1$". This says that $6$ is not a prime number, since $6=2\times 3$; this is an instance of the "iff" meaning in definitions. But it also says implicitly that $1$ is not a prime number, nor $-5$ nor $\pi$ nor $\exp(\pi\mathbf i/3)$ nor $\mathbf{GL}(3,\Bbb R)$, as none of these can be described as integers $p>1$; even a statement with "iff" would in itself not seem to state non-primality of those objects. But since it is a definition, anything that does not match its description is implicitly excluded from the property.

Without this implicit exclusion of cases not mentioned, it would be very hard indeed to give a complete definition of any property. Imagine (assuming the "everything is a set" philosophy) the ugliness of "a set $x$ is called a prime number if $x\in\Bbb Z$ and $x>1$ and ...".

As other pointed out, the "only if" part wouldn't make any sense as the process of defining the object is not over yet at the time you read the "only if".

But if it bothers you, you can always reformulate

An object $A$ is said to be new term if $P(A)$.

by

A new term is an object $A$ such that $P(A)$.

In your example, the reformulation would be :

A basis $\mathcal B$ of a topological space $T$ is a collection of open sets of $T$ such that every open set of $T$ is a union of elements of $\mathcal B$.

The deeper point here is that regardless whether we write "if" or "if and only if" in a definition, this is not the same as the usual meaning of "if and only if" as a material equivalence. The material equivalence of two propositions is the assertion that each one independently has a truth value, and the truth values are the same.

In a definition such as

a natural number $n$ is even if and only if $n$ is a multiple of $2$

we cannot read the "if and only if" as a material equivalence, because the left hand side does not have any truth value until after the definition is made. To view this definition as expressing an equivalence of two statements, we would need to already know that each side of the equivalence statement already has a truth value, but "$n$ is even" is (trivially) undefined before the definition is made.

Instead of expressing an equivalence, a definition tells us that the defined term is to be viewed as a synonym for the definition: a definition expresses an "is" relationship, not an "is equivalent to" relationship. For example, if I prove that $8$ is even, I do not need to apply modus ponens and the definition of an even number to conclude that $8$ is a multiple of $2$. Instead, I can directly "apply the definition": proving that "$8$ is even" already proves that "$8$ is a multiple of $2$", because the quoted phrases are synonyms for each other. One could also say that "$8$ is even" is an abbreviation of "$8$ is a multiple of $2$". In turn, that latter phrase is an abbreviation for "there is a natural number $m$ with $8=2m$".

In principle, one can remove all later definitions from an axiomatic theory, so that in the end one is left with synonymous statements that involve only the "undefined terms" of the theory.

For example, in Hilbert's axiomatic Euclidean plane geometry, there are only three undefined types of objects: point, lines, and planes. Every other statement, such as "the three angles in an equilateral triangle are all congruent", is in fact an abbreviation for a much longer statement about lines and points. This is not to say that the statement is equivalent to the longer statement about lines and points. The statement about angles has no truth value at all in this axiomatic framework except by virtue of its definition.

Yes. This is an unfortunate convention but is firmly established.