how to access array by index in a shell for loop in Makefile

Make expands the recipes once, just before passing the result to the shell. So, the recipe of your first attempt:

run:
    for m in 1 2 3 ; do \
        echo Info : MONTH = ${MONTH[$${m}]}; \
        echo Info : DATE  = ${DATE[$${m}]}; \
    done

Expands as:

for m in 1 2 3 ; do echo Info : MONTH = ; echo Info : DATE  = ; done

because there are no make variables named MONTH[${m}] or DATE[${m}]. The recipe of your second attempt:

run:
    for m in 1 2 3 ; do \
        echo Info : MONTH = $(word $$m, ${MONTH}); \
        echo Info : DATE  = $(word $$m, ${DATE}); \
    done

fails with error Makefile:5: invalid first argument to 'word' function: '$m'. Stop. when expanded by make because $m is not a valid value for the first parameter of make function word.

MONTH := jan feb mar
DATE := 01 03 05
MONTH.DATE := $(join $(addsuffix .,$(MONTH)),$(DATE))

MONTH := jan feb mar
DATE := 01 03 05
MONTH.DATE := $(join $(addsuffix .,$(MONTH)),$(DATE))
.PHONY: run
run:
    @for md in $(MONTH.DATE); do \
        m=$${md%.}; d=$${md#.}; \
        echo Info : MONTH = $$m; \
        echo Info : DATE = $$d; \
    done

MONTH := jan feb mar
DATE := 01 03 05
MONTH.DATE := $(join $(addsuffix .,$(MONTH)),$(DATE))
.PHONY: run $(MONTH.DATE)
run: $(MONTH.DATE)
$(MONTH.DATE):
    @md=$@; m=$${md%.}; d=$${md#*.}; \
    echo Info : MONTH = $$m; \
    echo Info : DATE = $$d

A side benefit is that make can process all jobs in parallel (try make -j3). If the jobs are computation intensive and your computer is multi-core the benefit can be significant.

MONTH := jan feb mar
DATE := 01 03 05
run: SHELL := /bin/bash
run:
    m=($(MONTH)) d=($(DATE)); \
    for i in {0..2}; do \
        echo Info : MONTH = $${m[i]}; \
        echo Info : DATE = $${d[i]}; \
    done