You can use

<<((?:(?!<<|>>).)*?\n(?s:.)*?)>>

See the regex demo. Details:

• << - a << string
• ((?:(?!<<|>>).)*?\n(?s:.)*?) - Group 1:
  • (?:(?!<<|>>).)*? - any zero or more chars (other than newline chars) that do not start >> or << char sequence, as few as possible
  • \n - a LF char
  • (?s:.)*? - any zero or more chars (including newline chars), as few as possible
• >> - a >> string

In your pattern <<(.*?\n*)*?>> you have a capture group and all parts are optional including the newline, so the non greedy quantifier *? can match until the first occurrence of >>

Also when repeating a capture group, the group value will hold the value of the last iteration, so instead you can put the capture group without a quantifier around the whole part that you want to capture.

If your strings start at the beginning of the line, you can use anchors and match at least a single line in between that does not start with either << or >>

^\s*<<(.*(?:\r?\n(?!<<|>>).*)+\r?\n)\s*>>$

Explanation

• ^ Start of string
• \s*<< Match optional leading whitspace chars and <<
• ( Capture group 1
  • .* Match the rest of the line
  • (?:\r?\n(?!<<|>>).*)+ Match a newline, and repeat at least 1 line not starting with << or >>
• \r?\n Match a newline
• ) Close group 1
• \s*>> Match optional leading whitspace chars and >>
• $ End of string

See a regex demo.

You can try this: <<[^>]*?\n[^>]*>>

Test regex here: https://regex101.com/r/vD3EgE/2

<<[^>]*?\n[^>]*>>

<<      match literal <<
[^>]*?  match any char that is not > as few as possible
\n      match a newline
[^>]*   match any char that is not > as few as possible
>>      match literal >>

• This will match a only if there is \n between << and >>.