I think you would need to do an eval of some type on $_ to make that happen, and that is BAD NEWS(tm)! You'll probably need to do some sort of regex substitution instead to keep it relatively safe.

Regex method worked, thanks

What I am missing here?

When they're in a file that isn't a perl program, and perl isn't executing that file, perl doesn't consider them perl code, so there is no interpolation that perl will do

What you've got there is called a template, so you need a template engine to do interpolation, like String::Interpolate/ String::Interpolate::RE , String::Defer, Interpolation, Template,Text::Xslate...

/startpattern/ (Ln 9) is NOT the same as /start pattern/ (in #start pattern in Ln 14.

Further, while others have mentioned the explanation for non-interpolation, no-one, OP included, has posted a solution -- so here's one (which omits the -- confusing to me -- script's use of both a file (of unspecified content) and __DATA__:

#start pattern
my name is $name
blahhhhhhh
my id is $id
blahhhh
#Endpattern

#newpattern
my name is $name
blash
my id is $id
#endofpattern

#startpattern
my name is $name
blaghhh
my id is $id
#endpattern

#start pattern
my name is fred
blahhhhhhh
my id is 42
blahhhh
#Endpattern

#1064933.txt
[download]

#!/usr/bin/perl
use 5.016;
use warnings;

# 1064933

my $name="guest";
my $id = 1;

$/ = "\n\n";

open(SOURCE, "<1064933.txt") or die "Can't open 1064933.txt, $!";
open(SINK, ">1064933OUT.txt") or die "Can't open 1064933OUT.txt, $!";
while (<SOURCE>) {
    if ( /start pattern/ .. /Endpattern/) {
        s/\$name/$name/;
        s/\$id/$id/;
        print SINK $_;
        next;
    }
}
[download]

D:\_Perl_\PMonks>cat 1064933OUT.txt
#start pattern
my name is guest
blahhhhhhh
my id is 1
blahhhh
#Endpattern

#start pattern
my name is fred
blahhhhhhh
my id is 42
blahhhh
#Endpattern
[download]

Note, however, that while this limits the exploitability of using an eval solution (by restricting the substitutions to the programmer's specs), it's far short of production-worthy. But then, the notion of globally replacing \$name with "guest" and making all \$id evaluate to 1 suggests that the example code has little resemblence to the actual task at hand.

What is next doing there? Also, /g might be needed if variables can be repeated on lines; and it gets very tricky if a variable's name is a part of other variable's name:

my car is $car_colour $car
[download]

لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ

Aaaarghhhhh. Good catch! Thank you.

next is a left-over (forgetfully 'unremoved') from an alternate approach which was later abandoned.

As to the possibility of multiple variables with similar name_parts -- all on a single line: that's not something I considered, as it appears to be outside the OP's spec. Oversight aside, however, that does indeed make the coding a little tricky...
      ... perhaps something like this [added to prev code and, yes, with the next removed :-)]:

say "\tDone with source and sink files\n";

$/ = "\n";      # restore record separator


my @choroba = ('choroba my car -- a $car -- is $car_colour. Choroba-',
              'choroba my car2 is $car_colour $car.  Choroba-',
              'choroba my car3 is $car_colour $car Choroba-',
              'choroba my car4 is $car_colour and |secondinstance: $ca
+r_colour| |firstinstance:$car| Choroba-',
              );

for (@choroba) {
    if ( $_ =~ /choroba/ .. /Choroba-/ ) {
        s/choroba/ /;           # ...
        s/Choroba-/ /;          # or set record separator
        s/\$car_colour/red/g;   # yup, need the g
        s/\$car[^_]{1}/ford/;
        say $_;
    }
}
[download]

which produces terminal output:

        Done with source and sink files

  my car -- a ford-- is red.
  my car2 is red ford
  my car3 is red ford
  my car4 is red and |secondinstance: red| |firstinstance:ford
[download]

Fun!