Using GNU find:
find . -type f -mtime +5 \
-regextype egrep -regex '.*[0-9]{4}-[0-9]{2}-[0-9]{2}[^/]*$' \
-delete
The regular expression will match any string in the basename of a pathname that contains a date on the form YYYY-MM-DD. Note that we may also match XXYYYXXYYYY-MM-DDZZ where XX and ZZ are some other characters.
The [^/]*$ at the end makes sure that we're actually matching the expression against the basename of the current pathname, and means "no / for the rest of the string, please".
Using a shell wildcard pattern instead (easier to maintain):
find . -type f -mtime +5 \
-name '*[0-9][0-9][0-9][0-9]-[01][0-9]-[0-3][0-9]*' \
-delete
Note that -mtime +5 is for files whose age as an integer number of days is strictly greater than 5, so 6 days and over. For files 5 days old or over, you'd need -mtime +4.