Yes, this is what bedmap is for, mapping or relating one set of BED elements to another:

$ bedmap --echo --echo-map-id --delim '\t' genes.bed windows.bed > answer.bed

The result:

$ more answer.bed
chr1    1400    1500    geneA    sample1;sample2
chr1    1900    2000    geneB    sample1;sample2

To explain the bedmap statement, the --echo option prints each gene from genes.bed. The --echo-map-id option prints the ID value of any mapped (overlapping) elements from windows.bed. The --delim option separates the gene and ID result with a tab character.

If you need the list of window IDs on separate lines, you can pipe the output to awk or other scripting tools to do this. Here is an untested script to get it into your desired format:

$ bedmap --echo --echo-map-id --delim '\t' genes.bed windows.bed \
   | awk '{ \
            n_ids = split($5, ids, ";"); \
            for (idx = 1; idx <= n_ids; idx++) { \
              print $1"\t"$2"\t"$3"\t"$4"\t"ids[idx]; \
            } \
          }' - \
   | sort-bed - \
   > split_answer.bed

The result:

$ more answer.bed
chr1    1400    1500    geneA    sample1
chr1    1400    1500    geneA    sample2
chr1    1900    2000    geneB    sample1
chr1    1900    2000    geneB    sample2

Note the use of sort-bed at the end of the pipeline, to ensure the output is a sorted BED file that you can use with other BEDOPS tools. If you don't care about the sort order of the output, you can remove the sort-bed call.

Piping avoids making intermediate files, which is time-consuming, and so it is a good way to deal with very big BED files. You might first test things out with a small subset of your input files to make sure the pipeline works, then run your full input files through the working pipeline.