It is possible for the first case

ClearAll@selectN
selectN[predicate_][n_ : Infinity] := Select[#, predicate, n] &

Names["System`*Values"] // 
 selectN[StringMatchQ[StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"]][]

Names["System`*Values"] // 
 selectN[StringMatchQ[StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"]][2]

You can use this to write select functions for different predicates e.g.

valuesSelect = 
  selectN[StringMatchQ[StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"]];
Names["System`*Values"] // valuesSelect[3]

unitDeterminantSelect = selectN[Det[#] == 1 &];
Tuples[{0, 1}, {3, 3}] // unitDeterminantSelect[3]

But if you need to be able to return all matches then maybe this is better

ClearAll@selectN
selectN[predicate_, n_ : Infinity] := Select[#, predicate, n] &

Names["System`*Values"] // 
 selectN[StringMatchQ[StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"]]

Names["System`*Values"] // 
 selectN[StringMatchQ[StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"], 2]

I get the result as shown below with ".*quater.*" as the regex:

In[29]:= EntityList["FiniteGroup"] // CanonicalName // 
 Select[StringMatchQ[First[#, #], RegularExpression[".*quater.*"], 
    IgnoreCase -> True] &]

Out[29]= {"Quaternion"}

I want to obtain the corresponding result in a more meaningful manner, say, as shown below for this case:

{"DirectProduct", {"Quaternion", {"CyclicGroup", 2}}}

I tried the following command, but failed:

In[55]:= EntityList["FiniteGroup"] // 
 Select[StringMatchQ[RegularExpression[".*space.*"], 
   IgnoreCase -> True]]

During evaluation of In[55]:= StringMatchQ::strse: String or list of strings expected at position 1 in StringMatchQ[RegularExpression[.*space.*],IgnoreCase->True][alternating group of degree 1].

During evaluation of In[55]:= StringMatchQ::strse: String or list of strings expected at position 1 in StringMatchQ[RegularExpression[.*space.*],IgnoreCase->True][alternating group of degree 2].

During evaluation of In[55]:= StringMatchQ::strse: String or list of strings expected at position 1 in StringMatchQ[RegularExpression[.*space.*],IgnoreCase->True][crystallographic point group C1].

During evaluation of In[55]:= General::stop: Further output of StringMatchQ::strse will be suppressed during this calculation.

Out[55]= {}

Thank you for your explanation. I still have another question as describe below.

The following form works:

In[28]:= Select[Names["System`*Values"], 
 StringMatchQ[
  StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"], 2]

Out[28]= {"DownValues", "OwnValues"}

But the following fails:

In[11]:= Names["System`*Values"] // 
 Select[StringMatchQ[
   StartOfString ~~ "Down" | "Own" | "Sub" | "Up" ~~ "Values"], 2]

During evaluation of In[11]:= StringMatchQ::argt: StringMatchQ called with 0 arguments; 1 or 2 arguments are expected.

Out[11]= StringMatchQ[][{"DefaultValues", "DownValues", 
  "DynamicModuleValues", "FormatValues", "NSolveValues", "NValues", 
  "OwnValues", "SingularValues", "SolveValues", "SubValues", 
  "SynthesizeMissingValues", "UpValues", "Values"}]

Thank you for your comment. Let me add a few words in response:

1. As for regex, this is just a minimal example I want to show. In fact, I want to some tips with the pattern like "(Down|Own|Sub|Up)Values".

2. The following results are equivalent:

   In[29]:= Map[Information][List[DownValues]] == 
    Map[Information, List[DownValues]]
   
   Out[29]= True
   

So, this implies that the following commands are equivalent:

Map[Information][List[DownValues]]
Map[Information, List[DownValues]]

But I can't find the document specification for the first usage mentioned above.