I have the same problem, which I normally solve manually.

But this could be solved with separate rules for = , =~ and => and/or "sections separations".

please note how this works

my $h_periods =
  {
    Mercury => { orbital => 0.24,  rotation => 58.64 },
    Venus   => { orbital => 0.62,  rotation => -243.02 },
    Earth   => { orbital => 1.00,  rotation => 1.00 },
    Mars    => { orbital => 1.88,  rotation => 1.03 },
    Jupiter => { orbital => 11.86, rotation => 0.41 },
    Saturn  => { orbital => 29.46, rotation => 0.43 },
    Uranus  => { orbital => 84.01, rotation => -0.72 },
    Neptune => { orbital => 164.8, rotation => 0.67 },
};
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That's because { is a section separator while ( is not

   `separate'  Each rule can define its own section separator, which
           describes how to identify the separation of "sections"
           within the region to be aligned.  Setting the `separate'
           attribute overrides the value of `align-region-separate'
           (see the documentation of that variable for possible
           values), and any separation argument passed to `align'.
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using M-x customize-group RET align RET will facilitate customizing (it provides a textual UI for lisp data)

the rule in question in align-rules-list is

 Alignment rule:
            Title: perl-assignment
            Required attributes:
            Regexp:
            (Regular expression to match)
            Choice: Value Menu Regexp: [^=!^&*-+<>/|     
]\(\s-*\)=[~>]?\(\s-*\)\([^>=     
]\|$\)
            Optional attributes:
            INS DEL Choice: Value Menu Paren group:
                        (Parenthesis group to use)
                        Choice: Value Menu Repeat:
                        INS DEL Integer: 1
                        INS DEL Integer: 2
                        INS
            INS DEL Choice: Value Menu Modes:
                        (Modes where this rule applies)
                        Lisp expression: align-perl-modes
            INS DEL Choice: Value Menu To Tab Stop:
                        (Should rule align to tab stops)
                        Boolean: Toggle  off (nil)
            INS
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this [~>]? part should be deleted or replaced.

(BTW: not sure what the -* is intended to do...)