You did use PREFIX=~ LIB=~/lib/perl5, right? It's normal to be in a "random" directory. It's actually the "arch" directory. A module (e.g. Foo::Bar) may install itself into

$LIB/Foo/Bar.*
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and/or

$LIB/$ARCH/Foo/Bar.*
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where $LIB is the value passed to LIB=, and $ARCH is some system-specific string.

In DBD::mysql's case, most if not all the files will be installed in the arch dir.

As long as PERL5LIB include $LIB, Perl will look into both directories.

$ echo $PERL5LIB
/home/ikegami/lib/perl5

$ perl -V
...
  @INC:
    /home/ikegami/lib/perl5/x86_64-linux-gnu-thread-multi
    /home/ikegami/lib/perl5
...
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Don't move files around, you're introducing problems!