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This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.

For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).

Let me also point out that Easton's theorem shows it is relatively consistent with ZFC that the GCH fails at every infinite regular cardinal, with $2^\kappa=\kappa^{++}$ for every regular $\kappa$. If we also chop the universe off at the first inaccessible cardinal, if there is one, then we will get a model of ZFC in which $\kappa^{<\kappa}$ is never equal to $\kappa$ for any uncountable cardinal $\kappa$. In such a model, there is not a single instance of $\text{card}(A)=\text{card}(B)$, to use your notation, except the countably infinite case, and the finite cases of size 0 and 1.

This hypothesis is refutable in ZFC for the case where $A$ has singular cardinality $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

Thus, we have provable counterexamples to the hypothesis, such as the case $A=\aleph_\omega$, for which the set $B$ will be strictly larger than $A$.

For other sets $A$, however, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (but still false for every singular cardinal).

Let me also mention that set theorists have produced models with a global violation of the GCH, meaning that $2^\kappa>\kappa^+$ for every infinite cardinal. [Thanks for comment of Andrés.] We can have, for example, $2^{\aleph_\alpha}=\aleph_{\alpha+2}$ for every $\alpha$. This situation requires large cardinal strength to produce. We may also assume without loss that there are no inaccessible cardinals, by chopping the universe off at the least inaccessible.

In such a model, we will have $\kappa<\kappa^{<\kappa}$ for every uncountable cardinal $\kappa$. This will be immediate for successor cardinals by the global failure of the GCH. It will hold for singular limits by the König's theorem observation above. And there will be no uncountable regular limit cardinals by our ommission of the inaccessibles.

In such a model, there is not a single instance of $\text{card}(A)=\text{card}(B)$, to use your notation, except the countably infinite case, and the finite cases of size 0 and 1.

This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.

For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).

Let me also point out that Easton's theorem shows it is relatively consistent with ZFC that the GCH fails at every infinite regular cardinal, with $2^\kappa=\kappa^{++}$ for every regular $\kappa$. If we also chop the universe off at the first inaccessible cardinal, if there is one, then we will get a model of ZFC in which $\kappa^{<\kappa}$ is never equal to $\kappa$ for any infinite cardinal $\kappa$. In such a model, there is not a single instance of $\text{card}(A)=\text{card}(B)$, to use your notation.

This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.

For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).

Let me also point out that Easton's theorem shows it is relatively consistent with ZFC that the GCH fails at every infinite regular cardinal, with $2^\kappa=\kappa^{++}$ for every regular $\kappa$. If we also chop the universe off at the first inaccessible cardinal, if there is one, then we will get a model of ZFC in which $\kappa^{<\kappa}$ is never equal to $\kappa$ for any uncountable cardinal $\kappa$. In such a model, there is not a single instance of $\text{card}(A)=\text{card}(B)$, to use your notation, except the countably infinite case, and the finite cases of size 0 and 1.

This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.

For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).

This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.

So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.

For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).

Let me also point out that Easton's theorem shows it is relatively consistent with ZFC that the GCH fails at every infinite regular cardinal, with $2^\kappa=\kappa^{++}$ for every regular $\kappa$. If we also chop the universe off at the first inaccessible cardinal, if there is one, then we will get a model of ZFC in which $\kappa^{<\kappa}$ is never equal to $\kappa$ for any infinite cardinal $\kappa$. In such a model, there is not a single instance of $\text{card}(A)=\text{card}(B)$, to use your notation.