m/=(?:"[^"']+")|(?:'[^"']+')/;
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As you see we check for both valid possibilities (hence the use of |). This means that we do not have to save what was the first quote.

Here is a full explanation (with thanks to YAPE::Regex::Explain):

The regular expression:

(?-imsx:=(?:"[^"']+")|(?:'[^"']+'))

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  =                        '='
----------------------------------------------------------------------
  (?:                      group, but do not capture:
----------------------------------------------------------------------
    "                        '"'
----------------------------------------------------------------------
    [^"']+                   any character except: '"', ''' (1 or
                             more times (matching the most amount
                             possible))
----------------------------------------------------------------------
    "                        '"'
----------------------------------------------------------------------
  )                        end of grouping
----------------------------------------------------------------------
 |                        OR
----------------------------------------------------------------------
  (?:                      group, but do not capture:
----------------------------------------------------------------------
    '                        '\''
----------------------------------------------------------------------
    [^"']+                   any character except: '"', ''' (1 or
                             more times (matching the most amount
                             possible))
----------------------------------------------------------------------
    '                        '\''
----------------------------------------------------------------------
  )                        end of grouping
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
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Please note this does not work with forms such as var1='"1"' or similar.

I thought about using $1 and $2, but that doesn't seem to work.

I'm not saying that this is the preferred solution... but just for general interest: in those cases, you could use \1 and \2 instead, e.g.

my $s = q(var1='1' var2="2" var3="'3'" var4='"4"');

while ($s =~ m/(\w+)=(["'])(.*?)\2/g) {
    print "(using quote $2): $1 = $3\n";
}
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Output:

(using quote '): var1 = 1
(using quote "): var2 = 2
(using quote "): var3 = '3'
(using quote '): var4 = "4"
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The ugly thing is that with input such as (incorrect according to your spec)

my $s = q(var1='1" var2="2');
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it would extract the entire '1" var2="2' substring as one single quoted value...

You'd have to work around that by disallowing some separator (like whitespace) within the quoted string, or some such, to properly group the assignments (e.g. using the char class [^\s] in place of .). That's problematic if you do need to allow spaces in the quoted values, though.

Update: fixed/simplified [^\2]+?  —> .*? in the regex, because on second thought, when looking at hipowls's suggestion below, it occurred to me that (a) the backref in the character class doesn't actually work :), (b) even if it did, that part of the regex would've been redundant anyway, due to the non-greedy match...

No, almut. It might look ugly, but it's the correct output. It might be a typo, but that's what we programmers have to deal with ;-)

---

s$$([},&%#}/&/]+}%&{})*;#$&&s&&$^X.($'^"%]=\&(|?*{%
+.+=%;.#_}\&"^"-+%*).}%:##%}={~=~:.")&e&&s""`$''`"e

Yes, it's correct from a purely syntactic point of view. Still, it's presumably not what the OP had in mind...

Anyhow, I just wanted to point out that there's a general disambiguation problem, in case the OP needs to allow arbitrary quoted content, which might contain the very separator that's used to split up the individual key-value expressions. It's unclear from the given spec, however, whether that's the case.

while (my ($var) = $line =~ s/^(\s*['"]+=//) {  # eat single var
  # eat a single quoted value and its trailing whitespace.
  $line =~ s/^'([^']+)'\s*// || $line =~ s/^"([^"]+)"\s*// ||
    die "invalid";  # or return, whatever you need
}
# if we haven't consumed all our input, this line is invalid.
die "invalid" if length $line;
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You could use a character class and a back reference instead of alternation.

my $string = qq{var1='1' var2="2" var3="3"};

while ( $string =~ /(\w+)=(['"])(.*?)\2/g) {
    print "$1 = $3\n";
}
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use 5.010_000;

while ( $string =~ /(?<variable>\w+)=(?<delim>['"])(?<value>.*?)\k<del
+im>/g) {
    say "$+{variable} = $+{value}";
}
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Note that when the string includes mixed quoting which is not valid in the OP's terms:

my $string = qq{var1='1' var2="2" var3="3" var4='4" var5="5' var6="correct"};

...hipowls non-5.10 version (5.10 not tested) produces "ugly" output in the sense used by almut and skeeve

var1 = 1
var2 = 2
var3 = 3
var4 = 4" var5="5
var6 = correct
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More significantly, if $str is in the form:

my $string = qq{var1='1' var2="2" var3="3" var4='4" var5="5'

...the output becomes:

var1 = 1
var2 = 2
var3 = 3
var4 = 4" var5=
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...which is the same as the output produced if var5 is properly quoted, var5='5':

my $string = qq{var1='1' var2="2" var3="3" var4='4" var5='5'};

i.e.,

var1 = 1
var2 = 2
var3 = 3
var4 = 4" var5=
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That sort of ambiguity may be a problem for OP.