Nothing but a newline is printed here (FreeBSD 12.1-STABLE r358002, perl 5.30.3, /bin/sh & zsh) for ...

$ perl -T -MDate::Manip::DM6 -E 'say $Date::Manip::VERSION'
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I get the same output -- 6.81 -- for both of ...

$ DATE_MANIP=DM5 perl -T -MDate::Manip -E 'say $Date::Manip::VERSION'

$ DATE_MANIP=DM6 perl -T -MDate::Manip -E 'say $Date::Manip::VERSION'
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... without any error messages. For me $ENV{ENV} is undefined in both shells; I have not explicitly (un)set it myself.

After explicitly making ENV an environment variable for the exercise, error messages are certainly reproducible. So something is stuffing ENV for you and possibly for OP. Wait ...

Ah. I see I did set ENV to shell configuration file path in long unused profile for Bourne shell (sh) & in bash{rc,_profile} for bash.