That's exactly what the join command is made for: it joins two files based on a common field:
$ awk '{print $1}' File1 | join - File2
M01605:153:000000000-B55NK:1:1101:10003:14536 2:N:0:1 GTTTGCGCCGATGTA
M01605:153:000000000-B55NK:1:1101:10007:14573 2:N:0:1 GGGGATAAGCGTTGC
M01605:153:000000000-B55NK:1:1101:10007:14573 2:N:0:1 GGGGATAAGCGTTGC
M01605:153:000000000-B55NK:1:1101:10009:9160 2:N:0:1 CAGAAGAGGTAATGT
M01605:153:000000000-B55NK:1:1101:10009:9160 2:N:0:1 CAGAAGAGGTAATGT
M01605:153:000000000-B55NK:1:1101:10009:9160 2:N:0:1 CAGAAGAGGTAATGT
M01605:153:000000000-B55NK:1:1101:10010:14111 2:N:0:1 CTGCGTACTGATAGC
M01605:153:000000000-B55NK:1:1101:10010:19939 2:N:0:1 TCCGTGGTGCCGGCA
M01605:153:000000000-B55NK:1:1101:10011:22802 1:N:0:1 TGAGTTCGGATAAAG
M01605:153:000000000-B55NK:1:1101:10011:22802 1:N:0:1 TGAGTTCGGATAAAG
Your files may be sorted numerically but not alphabetically, as expected by join. If join is complaining, slightly modify the command above to sort the input:
$ awk '{print $1}' File1 | sort | join - <(sort -k1,1 --stable File2)