• For "Is there any special bash reserved characters in bash regex expression?"
  The simple answer is yes.

• As for an example:

  url="https://www.youtube.com/playlist?list=PLmHVyfmcRKyxvxehq3fiGjKDsEyy6d4Tz"
  regex='^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list='
  
  idx=0
  (( idx++ )) # normal
  if [[ $url =~ ^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ )) # normal
  if [[ $url =~ $regex ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  yt='www\.youtube\.com\/'
  if [[ $url =~ ^https:\/\/${yt}playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ ^https:\/\/'${yt}'playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ '^https:\/\/'${yt}'playlist\?'(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  

• I guess its better to change the question:
  from
  "Is there any special bash reserved characters in bash regex expression?"
  to
  "How quotes work in bash regex expression? (Regarding special regex reserved character & special bash reserved characters.)"

• (Was expecting a way to write literal regex, that would be so much easier, seems not possible.)
  (see update below: use single quote ')

• For "Is there any special bash reserved characters in bash regex expression?"
  The simple answer is yes.

• As for an example:

  url="https://www.youtube.com/playlist?list=PLmHVyfmcRKyxvxehq3fiGjKDsEyy6d4Tz"
  regex='^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list='
  
  idx=0
  (( idx++ )) # normal
  if [[ $url =~ ^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ )) # normal
  if [[ $url =~ $regex ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  yt='www\.youtube\.com\/'
  if [[ $url =~ ^https:\/\/${yt}playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ ^https:\/\/'${yt}'playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ '^https:\/\/'${yt}'playlist\?'(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  

• For "Is there any special bash reserved characters in bash regex expression?"
  The simple answer is yes.

• As for an example:

  url="https://www.youtube.com/playlist?list=PLmHVyfmcRKyxvxehq3fiGjKDsEyy6d4Tz"
  regex='^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list='
  
  idx=0
  (( idx++ )) # normal
  if [[ $url =~ ^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ )) # normal
  if [[ $url =~ $regex ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  yt='www\.youtube\.com\/'
  if [[ $url =~ ^https:\/\/${yt}playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ ^https:\/\/'${yt}'playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ '^https:\/\/'${yt}'playlist\?'(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  

• I guess its better to change the question:
  from
  "Is there any special bash reserved characters in bash regex expression?"
  to
  "How quotes work in bash regex expression? (Regarding special regex reserved character & special bash reserved characters.)"

• (Was expecting a way to write literal regex, that would be so much easier, seems not possible.)

Update:

• For "Is there any special bash reserved characters in bash regex expression?"
  The simple answer is yes.

• As for an example:

  url="https://www.youtube.com/playlist?list=PLmHVyfmcRKyxvxehq3fiGjKDsEyy6d4Tz"
  regex='^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list='
  
  idx=0
  (( idx++ )) # normal
  if [[ $url =~ ^https:\/\/www\.youtube\.com\/playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ )) # normal
  if [[ $url =~ $regex ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  yt='www\.youtube\.com\/'
  if [[ $url =~ ^https:\/\/${yt}playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ ^https:\/\/'${yt}'playlist\?(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  (( idx++ ))
  if [[ $url =~ '^https:\/\/'${yt}'playlist\?'(.+&)?list= ]]; then echo "URL matches the regex - $idx"; fi
  

• I guess its better to change the question:
  from
  "Is there any special bash reserved characters in bash regex expression?"
  to
  "How quotes work in bash regex expression? (Regarding special regex reserved character & special bash reserved characters.)"

• (Was expecting a way to write literal regex, that would be so much easier, seems not possible.)
  (see update below: use single quote ')

Update:

• conclusion from comments & discussions:
  • use ' single quote & a variable outside the if statement.
  • here, ' in bash is actually analogous to / in js
    • -- so, the literal regex I mean (should have learnt more before asking.)
      (you may read comments & discussions below)
  • though, escaping single quote ' in bash -- '\'', is not as pretty as escaping / in js -- \/.)

js anology:

const str = "The pizza is 2' and 100$price";

if (/[0-9]+' and [0-9]+\$price/.test(str)) {
  console.log('literal regex');
}
if (new RegExp("[0-9]+' and [0-9]+\\$price").test(str)) {
  console.log('non literal regex');
}