I mean, you are welcome to try this, via biomaRt. I admit to not fully understanding how Ensembl have determined these mappings 'behind the scenes', but I imagine and hope that they are based on genome alignments and / or general sequence homology, following from GenoMax's point.

This is an elaboration of an answer that I gave on Bioconductor: support.bioconductor.org/p/132551/#132568

Setup

require(biomaRt)

listDatasets(useMart('ensembl'))

datasets <- listDatasets(useMart('ensembl'))
datasets[grep('mmulatta', datasets[,1]),]
                  dataset             description version
104 mmulatta_gene_ensembl Macaque genes (Mmul_10) Mmul_10

rhesus <- useMart('ensembl', dataset="mmulatta_gene_ensembl")
human <- useMart('ensembl', dataset="hsapiens_gene_ensembl")

Create a Rhesus macaque (M. mulatta) annotation lookup table (not necessary)

table <- getBM(
  attributes = c('ensembl_gene_id', 'vgnc', 'vgnc_trans_name', 'external_gene_name'),
  mart = rhesus)
head(table[table$external_gene_name != '',], 30)

      ensembl_gene_id        vgnc vgnc_trans_name external_gene_name
2  ENSMMUG00000036181                                             U6
3  ENSMMUG00000000634 VGNC:100195      ZNF692-201             ZNF692
4  ENSMMUG00000000634 VGNC:100195      ZNF692-202             ZNF692
5  ENSMMUG00000000634 VGNC:100195      ZNF692-203             ZNF692
6  ENSMMUG00000037875                                         ZNF672
7  ENSMMUG00000000632  VGNC:77334     SH3BP5L-201            SH3BP5L
8  ENSMMUG00000000632  VGNC:77334     SH3BP5L-202            SH3BP5L
9  ENSMMUG00000000632  VGNC:77334     SH3BP5L-203            SH3BP5L
14 ENSMMUG00000031199                                         OR2T27
17 ENSMMUG00000025700                                          Y_RNA
18 ENSMMUG00000049101                                         OR2T10
21 ENSMMUG00000055842                                         OR14I1
23 ENSMMUG00000062331                                          OR9G1
26 ENSMMUG00000005566                                          OR2T1
27 ENSMMUG00000064874                                          OR2T6
29 ENSMMUG00000038524                                          OR2M4
35 ENSMMUG00000062908                                          Y_RNA
36 ENSMMUG00000057748                                          OR2L2
37 ENSMMUG00000051218                                          Y_RNA
38 ENSMMUG00000050041                                          OR2L5
40 ENSMMUG00000054556                                          Y_RNA
43 ENSMMUG00000059751                                          Y_RNA
46 ENSMMUG00000038159                                          OR2W3
47 ENSMMUG00000005570 VGNC:100177      TRIM58-201             TRIM58
48 ENSMMUG00000006214                                         OR11L1
50 ENSMMUG00000049478                                         OR9H1P
51 ENSMMUG00000062849                                          OR1C1
52 ENSMMUG00000064145                                         OR14K1
53 ENSMMUG00000059455                                         OR14A2
54 ENSMMUG00000016175                                          OR6F1

Create a merged annotation table between Rhesus macaque (M. mulatta) and Human (H. sapiens)

table <- getLDS(
  mart = rhesus,
  attributes = c('ensembl_gene_id', 'vgnc', 'external_gene_name', 'chromosome_name'),
  martL = human,
  attributesL = c('ensembl_gene_id','hgnc_symbol','gene_biotype', 'chromosome_name'))

head(table)
      Gene.stable.ID VGNC.ID Gene.name Chromosome.scaffold.name
1 ENSMMUG00000052610                                          1
2 ENSMMUG00000065379              ND4L                       MT
3 ENSMMUG00000060150           5S_rRNA           QNVO02001753.1
4 ENSMMUG00000065383              ATP8                       MT
5 ENSMMUG00000063555                                          8
6 ENSMMUG00000065366              COX3                       MT
  Gene.stable.ID.1 HGNC.symbol      Gene.type Chromosome.scaffold.name.1
1  ENSG00000271254             protein_coding                 KI270711.1
2  ENSG00000212907     MT-ND4L protein_coding                         MT
3  ENSG00000278457                       rRNA                 KI270442.1
4  ENSG00000228253     MT-ATP8 protein_coding                         MT
5  ENSG00000198695      MT-ND6 protein_coding                         MT
6  ENSG00000198938      MT-CO3 protein_coding                         MT

dim(table)
[1] 23385     8

table[6000:6010,]
         Gene.stable.ID    VGNC.ID    Gene.name Chromosome.scaffold.name
6000 ENSMMUG00000007970                                                3
6001 ENSMMUG00000005091 VGNC:75524        NRBP1                       13
6002 ENSMMUG00000010505 VGNC:82153       GTF3C2                       13
6003 ENSMMUG00000009610 VGNC:81776      TMEM196                        3
6004 ENSMMUG00000061977                  SPIRE1                       18
6005 ENSMMUG00000000452                    LSM7                       19
6006 ENSMMUG00000003962 VGNC:76941         RRM1                       14
6007 ENSMMUG00000014165                   CHADL                       10
6008 ENSMMUG00000009818 VGNC:75393          NPY                        3
6009 ENSMMUG00000026995            mml-mir-320a                        8
6010 ENSMMUG00000003815 VGNC:69540     ADAMTSL5                       19
     Gene.stable.ID.1 HGNC.symbol      Gene.type Chromosome.scaffold.name.1
6000  ENSG00000146574       CCZ1B protein_coding                          7
6001  ENSG00000115216       NRBP1 protein_coding                          2
6002  ENSG00000115207      GTF3C2 protein_coding                          2
6003  ENSG00000173452     TMEM196 protein_coding                          7
6004  ENSG00000134278      SPIRE1 protein_coding                         18
6005  ENSG00000130332        LSM7 protein_coding                         19
6006  ENSG00000167325        RRM1 protein_coding                         11
6007  ENSG00000100399       CHADL protein_coding                         22
6008  ENSG00000122585         NPY protein_coding                          7
6009  ENSG00000208037     MIR320A          miRNA                          8
6010  ENSG00000185761    ADAMTSL5 protein_coding                         19



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