APL(Dyalog Unicode), ^{118 114 90 88 83} 80 bytes ^SBCS

{⊃,/⍪⌿⍵{' '⍺[⍵]}¨1+(⍺ ⍺∘⍴¨o(⌽o←∘.=⍨⍳⍺)i(⌽⍺/⍺↑1)(⍉⍺ ⍺⍴i←(⌈⍺÷2)=⍳⍺)0)['\/|_- '⍳⍵]} 

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A dfn submission which takes a char matrix on the right and a scale factor on the left.

Indexes each character into its specific pattern, and then uses the original matrix to index back into it.

More byte saves can be done by figuring out a formula for getting the boolean arrays.

V (vim), 311 bytes

"aDj:%s:\([ \-_]\):=repeat("\\1",a) :g :%s:\\:\\=repeat(" ",a-1) :g :%s:/:=repeat(" ",a-1) /:g :%s:|:=repeat(" ",a/2) |=repeat(" ",a/2) :g {qqYp:s:\\ : \\:ge :s: /:/ :ge qdkG{Go ggqtjjkk@a@qdd@tq@tggqb@ajk:s:_:Y:ge j@bq@bi=a/2+1 D@"kkqc:s:-:X:ge @ak@cq@c:%s:[-_]: :g :%s:X:-:g :%s:Y:_:g gg:s:-: :g 

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Thanks to PyGamer for the idea of solving this in Vim. It's not extremely well golfed, can change some substitutions and bad hacks around the many macros that are in use.

Outputs the required art with two trailing newlines.

C (clang), 170 147 143 137 bytes

a,j,i;f(*m,x,z){for(;*m;m+=x)for(j=z;j--;puts(""))for(i=-1;++i<x*z;putchar((a-92?a-47?a-45?a-95|j:j-~j-z:i%z-j:~j+z-i%z)?32:a))a=m[i/z];} 

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• thanks to @ceilingcat suggestion !
• added for loop instead of recursion
• thanks to @Johan du Toit for suggesting using int*array as input!

f(*m,x,z){ function taking:

• m : char int array without newlines
• x : width
• z : scale

m+=x;*m&&f(m,x,z); => for(;*m;m+=x)
for every row of input, loop actually better than recursion.

for(j=z;j--;puts(""))
we iterate z times every row of input and put \n each time.

for(i=-1;++i<x*z;putchar(..) )
we put x*z characters.

a=m[i/z]
we use a to save on m[i/z] repetitions which is current input being enlarged.

putchar(( ... )?32:a
... => many nested ternary operators to select proper character based on a and i / j relations which evaluates to 1 or 0 , we then puts a space or a.

J, 89 bytes

[(a{~[:,/[:,./"3[*({0:0}(,:|.)@=@i.,/@,~(1,>.@-:)(,:~|:)@|."{1:0},~$0:))~(a=.' _|-\/')i.] 

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K (oK), 261 184 165 163 159 bytes

{k:y;v:" ";a:{[g]{(x#v),g,(k-x+1)#v}'!k};h:{[s]{k#s@x=_k%2}'!k};,/{,/'+{$[92=x;a x;47=x;|:'a x;45=x;h v,x;95=x;{k#" _"@x=k-1}'!k;v=x;(k;k)#v;+h v,x]}'x}'";"\x} 

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I lost my sanity several times while creating this. Thanks to ngn and Razetime for helping me with this.

Thanks to Razetime for helping me to golf off a lot.

The basic approach is to split by newlines, replace each character with a list of subrows, take the transpose of that, join each line, and flatten.

K (ngn/k), 51 bytes

{,/,'/'`c$32|x*(a@<a;+a;|=y;=y;a=:|a:y#'!y)8!19!-x} 

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Python 2, 135 bytes

Takes input as a list of lines.

s,f=input() R=range(f) for r in s: for w in R:print''.join([i-w,i-~w-f,w-f/2,f+~w,i-f/2][ord(c)%23%5]and' 'or c for c in r for i in R) 

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Charcoal, 55 bytes

Ｎθ≔⊘⊕θη↑ＷＳ«Ｊ⁰⁺θⅉＦι¿⁼κ_×θκ«Ｍ⊖η↗≡κ-Ｐ-η/Ｐ/η|Ｐ|η\Ｐ\ηＰκＭ⊖η↘→ 

Try it online! Link is to verbose version of code. Explanation:

Ｎθ≔⊘⊕θη 

Input the scale factor n and calculate half, rounded up.

↑ＷＳ«Ｊ⁰⁺θⅉ 

Output each transformed row on its own set of n lines.

Ｆι 

Loop through the characters on each line.

¿⁼κ_×θκ« 

If the next character is a _ then print n of them, otherwise...

Ｍ⊖η↗ 

Move to the centre of the square.

≡κ-Ｐ-η/Ｐ/η|Ｐ|η\Ｐ\ηＰκ 

Print a line in the desired direction. (TIO's version of Charcoal doesn't support computed multidirectionals, so I'm not sure whether that would beat a switch statement.)

Ｍ⊖η↘→ 

Move to the bottom left of the next square.

JavaScript (ES2020), 157 151 148 147 bytes

Takes a 2D character array A and a size n as f(A)(n).

A=>n=>A.flatMap(L=>(N=[...Array(n)]).map((_,y)=>L.flatMap(c=>N.map((_,x)=>[a=y-n+1,a+y,a+=x,a+x-y]['_-/|'.indexOf(c)]??x-y?' ':c)).join``)).join` ` 

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JavaScript (ES2020), 157 151 148 147 bytes

A=>n=>A.flatMap(L=>(N=[...Array(n)]).map((_,y)=>L.flatMap(c=>N.map((_,x)=>({_:a=y-n+1,'-':a+y,'/':a+=x,'|':a+x-y})[c]??x-y?' ':c)).join``)).join` ` 

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JavaScript (ES2019), 161 158 152 149 148 bytes

A=>n=>A.flatMap(L=>(N=[...Array(n)]).map((_,y)=>L.flatMap(c=>N.map((_,x)=>[a=y-n+1,x-y,a+y,a+=x,a+x-y]['_\\-/|'.indexOf(c)]?' ':c)).join``)).join` ` 

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Haskell, 97 bytes

n%a|z<-[1-n,3-n..n]=[do c<-r;x<-z;max" "[c|(l,0)<-zip"|_/\\-"[x,y-n+1,y+x,y-x,y],c==l]|r<-a,y<-z] 

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I/O: size%listOfLines is a new list of lines.

PowerShell, 162 152 149 143 138 bytes

param($k,$o)$o|%{$l=$_;1..$k|%{$y=$_ -join($l|% t*y|%{$c=$_;1..$k|%{" $c"[@{1=$y-$_+$k+1 8=$_+$y 7=2*$_ 6=2*$y 4=$y+1}[$c%13]-eq$k+1]}})}} 

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Less golfed:

param($k,$origLines) $origLines|%{ $line = $_ 1..$k|%{ $y = $_ -join($line|% toCharArray|%{ $char = $_ 1..$k|%{ $x = $_ $rasterFont = @{ 1 = $y-$x+$k+1 # \ 8 = $x+$y # / 7 = 2*$x # | 6 = 2*$y # - 4 = $y+1 # _ } " $char"[$rasterFont[$char%13]-eq$k+1] # space or $char } }) } }