Unix Parse Varying Named Value into seperate rows

Could you please try following, written and tested with shown samples in GNU awk with new version of it. Since OP's awk version is old so if anyone having old version of awk then try changing it to awk --re-interval

awk '
BEGIN{
  FS=OFS="|"
}
FNR==1{ next }
{
  first=$1
  while(match($0,/(name[0-9]+;?){1,}=(value[0-9]+)?/)){
    print first,substr($0,RSTART,RLENGTH)
    $0=substr($0,RSTART+RLENGTH)
  }
}'  Input_file

Output will be as follows.

1|name1=value1
1|name3;name4=value2
1|name5=value5
2|name1=value1
2|name2=value2
2|name6=
2|name7=value7
2|name8=value8

Explanation: Adding detailed explanation for above(following is for explanation purposes only).

awk '                                        ##Starting awk program from here.
BEGIN{                                       ##Starting BEGIN section from here.
  FS=OFS="|"                                 ##Setting FS and OFS wiht | here.
}
FNR==1{ next }                               ##If line is first line then go next, do not print anything.
{
  first=$1                                   ##Creating first and setting as first field here.
  while(match($0,/(name[0-9]+;?){1,}=(value[0-9]+)?/)){
##Running while loop which has match which has a regex of matching name and value all mentioned permutations and combinations.
    print first,substr($0,RSTART,RLENGTH)    ##Printing first and sub string(currently matched one)
    $0=substr($0,RSTART+RLENGTH)             ##Saving rest of the line into current line.
  }
}' Input_file                                ##Mentioning Input_file name here.

Sample data:

$ cat name.dat
ID|Text
1|name1=value1;name3;name4=value2;name5=value5
2|name1=value1;name2=value2;name6=;name7=value7;name8=value8

One awk solution:

awk -F"[|;]" '                                                           # use "|" and ";" as input field delimiters
FNR==1 { next }                                                          # skip header line
       { pfx=$1 "|"                                                      # set output prefix to field 1 + "|"
         printpfx=1                                                      # set flag to print prefix

         for ( i=2 ; i<=NF ; i++ )                                       # for fields 2 to NF
             {
               if ( printpfx)     { printf "%s",   pfx  ; printpfx=0 }   # if print flag == 1 then print prefix and clear flag
               if ( $(i)  ~ /=/ ) { printf "%s\n", $(i) ; printpfx=1 }   # if current field contains "=" then print it, end this line of output, reset print flag == 1
               if ( $(i) !~ /=/ ) { printf "%s;",  $(i) }                # if current field does not contain "=" then print it and include a ";" suffix
             }
       }
' name.dat

The above generates:

1|name1=value1
1|name3;name4=value2
1|name5=value5
2|name1=value1
2|name2=value2
2|name6=
2|name7=value7
2|name8=value8

A Bash solution:

#!/usr/bin/env bash

while IFS=\| read -r id text || [ -n "$id" ]; do
  IFS=\; read -r -a kv_arr < <(printf %s "$text")
  printf "$id|%s\\n" "${kv_arr[@]}"
done < <(tail -n +2 a.txt)

A plain POSIX shell solution:

#!/usr/bin/env sh

# Chop the header line from the input file
tail -n +2 a.txt |
# While reading id and text Fields Separated by vertical bar
while IFS=\| read -r id text || [ -n "$id" ]; do
  # Sets the separator to a semicolon
  IFS=\;
  # Print each semicolon separated field formatted on
  # its own line with the ID
  # shellcheck disable=SC2086 # Explicit split on semicolon
  printf "$id|%s\\n" $text
done

Input a.txt:

ID|Text
1|name1=value1;name3;name4=value2;name5=value5
2|name1=value1;name2=value2;name6=;name7=value7;name8=value8

Output:

1|name1=value1
1|name3
1|name4=value2
1|name5=value5
2|name1=value1
2|name2=value2
2|name6=
2|name7=value7
2|name8=value8

You have some good answers and an accepted one already. Here is a much shorter gnu awk command that can also do the job:

awk -F '|' 'NR > 1 {
   for (s=$2; match(s, /([^=]+=[^;]*)(;|$)/, m); s=substr(s, RLENGTH+1))
      print $1 FS m[1]      
}' file.txt

1|name1=value1
1|name3;name4=value2
1|name5=value5
2|name1=value1
2|name2=value2
2|name6=
2|name7=value7
2|name8=value8