4 added 312 characters in body
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Wolfram Language (Mathematica), 5652 bytes

Max[Length/@Select[Subsets@Divisors[s=#]@Cases[Subsets@Divisors@#,Times@@#==s&]]&{a__}/;1a==#]]&

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4-bytes saved thanks to @attinat

Here is also a 153 bytes version that calculates 1099511627776 and 10^15

Max[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length@s,DuplicateFreeQ],5!]]+1&      

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The result for 10^15 is 12

{1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}

Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 153 bytes version that calculates 1099511627776 and 10^15

Max[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length@s,DuplicateFreeQ],5!]]+1&      

Try it online!

The result for 10^15 is 12

{1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}

Wolfram Language (Mathematica), 52 bytes

Max[Length/@Cases[Subsets@Divisors@#,{a__}/;1a==#]]&

Try it online!

4-bytes saved thanks to @attinat

Here is also a 153 bytes version that calculates 1099511627776 and 10^15

Max[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length@s,DuplicateFreeQ],5!]]+1&      

Try it online!

The result for 10^15 is 12

{1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}

3 added 153 characters in body
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Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 123153 bytes version that calculates 1099511627776 inand 1 sec10^15

Length@Last@Select[Times@@@TakeList[sMax[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length[s=Flatten[Table@@@FactorInteger[#]]]@IntegerPartitions@Length@s,DuplicateFreeQ]+1&DuplicateFreeQ],5!]]+1&      


Try it online!Try it online!

The result for 10^15 is 12

{1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}

Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 123 bytes version that calculates 1099511627776 in 1 sec

Length@Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length[s=Flatten[Table@@@FactorInteger[#]]],DuplicateFreeQ]+1&     


Try it online!

Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 153 bytes version that calculates 1099511627776 and 10^15

Max[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length@s,DuplicateFreeQ],5!]]+1&      

Try it online!

The result for 10^15 is 12

{1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}

2 added 542 characters in body
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Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 123 bytes version that calculates 1099511627776 in 1 sec

Length@Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length[s=Flatten[Table@@@FactorInteger[#]]],DuplicateFreeQ]+1&     


Try it online!

Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Wolfram Language (Mathematica), 56 bytes

Max[Length/@Select[Subsets@Divisors[s=#],Times@@#==s&]]&

Try it online!

Here is also a 123 bytes version that calculates 1099511627776 in 1 sec

Length@Last@Select[Times@@@TakeList[s,#]&/@IntegerPartitions@Length[s=Flatten[Table@@@FactorInteger[#]]],DuplicateFreeQ]+1&     


Try it online!

1
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