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In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: $\sqsubseteq$) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar): $$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge z\mathrel{\bot} y) $$ where $z\mathrel{\bot} y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq z)$ ($z$ and $y$ are incompatible, otherwise they are compatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence:

In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: $\sqsubseteq$) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar): $$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge z\mathrel{\bot} y) $$ where $z\mathrel{\bot} y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq y)$ ($z$ and $y$ are incompatible, otherwise they are compatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence:

I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP question, Andreas Blass answer and Carl Mummert comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory.

Defining proper part as $x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y$ we may define mereological atoms (or points, if you prefer the name): $$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$ Now, in case $a_1,\ldots,a_n$ are atoms we can indeed treat $\bigl[a_1,\ldots,a_n\bigr]$ as a counterpart of $\{a_1,\ldots,a_n\}$ (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass: $$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$ works fine. But it does not work for example for: $$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$ since under the interpretation in question for every $a_i$: $$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$ Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.)

I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP's question, Andreas Blass's answer and Carl Mummert's comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory.

Defining proper part as $x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y$ we may define mereological atoms (or points, if you prefer the name) as objects minimal w.r.t. $\sqsubset$: $$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$ Now, in case $a_1,\ldots,a_n$ are atoms we can indeed treat $\bigl[a_1,\ldots,a_n\bigr]$ as a counterpart of $\{a_1,\ldots,a_n\}$ (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass: $$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$ works fine. But it does not work for example for: $$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$ since under the interpretation in question for every $a_i$: $$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$ Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.)

To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set): $$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$ while its mereological counterpart is usually not true. That is it is the case that: $$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$ but is NOT the case that: $$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x). $$ It may be interesting (in my opinion) to consider the system of mereology with the implication above taken as an axiom. This could shed some light on the nature of difference between both understandings of sets.

To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set): $$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$ while its mereological counterpart is usually not true. That is it is the case that: $$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$ but is NOT the case that: $$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x). $$

EDIT: Originally I suggested that it might be interesting to consider a system of mereology with the implication above taken as an axiom. However, in the comment below Andreas pointed to the fact that this entails linearity of $\sqsubseteq$. The consequence is that the class of models of the theory which consists of poset axioms+separation+existence of mereological sums narrows down to one-element (up to isomorphism) class, the only model being degenerate one-element structure.