So, why does "eq" work like this

Because the internal representation of the two numbers is the same. Just because your code uses different formats to specify them does not affect how they are stored internally. Try this version of your code to see the values:

perl -we '$a = 1e-3; $b = 0.001; if ( $a eq $b ) { print "$a and $b ar
+e equal.\n" }'
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BTW, try to avoid $a and $b for arbitrary variables in real code because they have special meanings for sorting.

The stringified form also depends if the values began as strings or numbers.

$x = '1e-3'; 
$y = 0.001; 
if ( $x eq $y ) { print "$x and $y are equal.\n" }
else { print "$x and $y are not equal" }

print "\n";

$x = $x + 0;
if ( $x eq $y ) { print "$x and $y are equal.\n" }
else { print "$x and $y are not equal" }
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Result is:

1e-3 and 0.001 are not equal
0.001 and 0.001 are equal.
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In the first comparison, $x is a string. In the second comparison, $x is numeric before it is stringified for comparison.

The Scalar::Util documentation is useful here. https://perldoc.perl.org/Scalar/Util.html#isdual

$a = 1e-3; $b = 0.001; if ( $a eq $b ) { ...

Note that this doesn't really have anything to do with eq or ==, but with how Perl handles numeric literals in its source code - they are converted to Perl's internal representation first.

# force numbers to strings before printing:
$ perl -wMstrict -le 'print "".1e-3; print "".0.001;
    print "".0.0_0_1; print "".0x1.0624dd2f1a9fcp-10'
0.001
0.001
0.001
0.001
# edited for brevity:
$ perl -wMstrict -MDevel::Peek -le 'Dump( 1e-3 )'
  NV = 0.001
$ perl -wMstrict -MDevel::Peek -le 'Dump( 0.001 )'
  NV = 0.001
$ perl -wMstrict -MDevel::Peek -le 'Dump( 0.0_0_1 )'
  NV = 0.001
$ perl -wMstrict -MDevel::Peek -le 'Dump( 0x1.0624dd2f1a9fcp-10 )'
  NV = 0.001
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would two numbers always numerically oompared by "eq"

If I take the question to mean "does eq do numeric comparisons automatically when presented with numbers", the answer is no, it always converts its operands to strings first (I'm ignoring overloading here).

can I rely on this in general

But my question would be: Why would you want to rely on this? If you want to do a numeric comparison, you should use the numeric == (always keeping in mind the limitations of floating-point numbers!*), and if you want to know if two values converted to strings are equal, use eq - and looking at the above, eq's behavior makes sense here.

At the moment, I can't think of a case where one would want to do such an "automatic numeric comparison". Could this be an XY problem - are you perhaps trying to do something with user input? (Update: If you want a number to retain its formatting, you need to store it as a string instead of a number.)

* Update: And if you want to know whether two numbers are approximately equal:

$ perl -wMstrict -le '
    if ( sprintf("%.3f",0.001) eq sprintf("%.3f",0.001001) )
      { print "yes" } else { print "no" }'
yes
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$ perl -le 'print "ok" if 1.4142135623731 eq 1.4142135623730951;'
ok
$ perl -le 'print "ok" if 1.4142135623731 != 1.4142135623730951;'
ok
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