#Java 8, 100 84 78 bytes

a->b->{for(int i=0;;)System.out.println(a.split("\n")[i]+b.split("\n")[i++]);}

Exits with an ArrayIndexOutOfBoundsException to STDERR after it has printed the result to STDOUT, which is allowed.

-6 bytes thanks to @OlivierGrégoire.

Explanation:

Try it online.

a->b->{                        // Method with two String parameters and no return-type
  for(int i=0;;)               //  Loop over the substrings of the first input
    System.out.println(        //   Print:
     a.split("\n")[i]          //    The substring of the first input
     +b.split("\n")[i++]);}    //    plus the same-indexed substring of the second input

Java 8, 100 84 78 bytes

a->b->{for(int i=0;;)System.out.println(a.split("\n")[i]+b.split("\n")[i++]);}

Exits with an ArrayIndexOutOfBoundsException to STDERR after it has printed the result to STDOUT, which is allowed.

-6 bytes thanks to @OlivierGrégoire.

Explanation:

Try it online.

a->b->{                        // Method with two String parameters and no return-type
  for(int i=0;;)               //  Loop over the substrings of the first input
    System.out.println(        //   Print:
     a.split("\n")[i]          //    The substring of the first input
     +b.split("\n")[i++]);}    //    plus the same-indexed substring of the second input

#Java 8, 100 84 bytes

a->b->{int i=0;for(String t:a.split("\n"))System.out.println(t+b.split("\n")[i++]);}

Try it online.

Explanation:

a->b->{                        // Method with two String parameters and no return-type
  int i=0;                     //  Index-integer, starting at 0
  for(String t:a.split("\n"))  //  Loop over the substrings of the first input
    System.out.println(t       //   Print this substring
     +b.split("\n")[i++]);}    //   plus the same-indexed substring of the second input

#Java 8, 100 84 78 bytes

a->b->{for(int i=0;;)System.out.println(a.split("\n")[i]+b.split("\n")[i++]);}

Exits with an ArrayIndexOutOfBoundsException to STDERR after it has printed the result to STDOUT, which is allowed.

-6 bytes thanks to @OlivierGrégoire.

Explanation:

Try it online.

a->b->{                        // Method with two String parameters and no return-type
  for(int i=0;;)               //  Loop over the substrings of the first input
    System.out.println(        //   Print:
     a.split("\n")[i]          //    The substring of the first input
     +b.split("\n")[i++]);}    //    plus the same-indexed substring of the second input

#Java 8, 100 bytes

a->b->{for(int i=0;i<a.split("\n").length;System.out.println(a.split("\n")[i]+b.split("\n")[i++]));}

Try it online.

Alternative of equal size (100 bytes):

a->b->{String[]t=a.split("\n");for(int i=0;i<t.length;System.out.println(t[i]+b.split("\n")[i++]));}

Try it online.

Explanation:

a->b->{  // Method with two String parameters and String return-type
  for(int i=0;i<a.split("\n").length;
         //  Loop `n` times, where `n` is the amount of "\n"
    System.out.println(a.split("\n")[i]+b.split("\n")[i++]));}
         //   Print both substrings concatted with each other, plus a new-line

#Java 8, 100 84 bytes

a->b->{int i=0;for(String t:a.split("\n"))System.out.println(t+b.split("\n")[i++]);}

Try it online.

Explanation:

a->b->{                        // Method with two String parameters and no return-type
  int i=0;                     //  Index-integer, starting at 0
  for(String t:a.split("\n"))  //  Loop over the substrings of the first input
    System.out.println(t       //   Print this substring
     +b.split("\n")[i++]);}    //   plus the same-indexed substring of the second input