Does perl have something analogous to the shell's source?

Not for shell code, as shell code needs to be interpreted by a shell (likewise, a shell cannot run Perl code).

As for "the shell which the Perl script is running in", Perl is started from a shell, but it's a separate subprocess, so strictyl speaking, the Perl script is not running "in" a shell.  This means the same child-parent restrictions apply, i.e. the Perl script (nor any further child processes started from it) cannot set the environment of its parent shell.

Update: maybe this clarifies it somewhat. When you run something from the command line, for example

$ perl -e 'system "ps Tf ; sleep 1"'
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you see child-parent relationships as follows (indenting means "child of")

  PID TTY      STAT   TIME COMMAND
29353 pts/7    S      0:00 bash                                   # th
+e initial interactive command shell
29388 pts/7    S+     0:00  \_ perl -e system "ps Tf ; sleep 1"   # yo
+ur Perl script
29389 pts/7    S+     0:00      \_ sh -c ps Tf ; sleep 1          # a 
+shell started by system()
29390 pts/7    R+     0:00          \_ ps Tf                      # a 
+subprocess run by the shell
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All those are separate processes created via fork/exec, and no child is able to manipulate the environment of its respective parent process. But it does inherit the environment of its parent.