Reads are compatible for aligners like BWA, bowtie2, etc (I tested),  even if the /1, /2 suffix not exists.
You need to be caution, make sure the first part of name in read1 and read2 are identical, and in the same order. 
What you paste in the post does not like the first two headers, the order is not correct.
I checked the fastq files from NCBI-SRA and EBI-ENA for the first two read name in read1 and read2
EBI-ENA version
As @genomax pointed out in EBI-ENA, ERR2184190_1.fastq.gz,   ERR2184190_2.fastq.gz 
found /1 and /2 suffix in tail of read name
==> read1.fq <==
@ERR2184190.1 SBS123:200:C3PFWACXX:6:1101:1462:1956/1
@ERR2184190.2 SBS123:200:C3PFWACXX:6:1101:1487:1970/1
==> read2.fq <==
@ERR2184190.1 SBS123:200:C3PFWACXX:6:1101:1462:1956/2
@ERR2184190.2 SBS123:200:C3PFWACXX:6:1101:1487:1970/2
SRA-toolkit download version
not found /1, /2 suffix
$ prefetch ERR2184190
$ fasterq-dump --threads 8 --split-3 ERR2184190.sra
==> read1.fq <==
@ERR2184190.sra.1 SBS123:200:C3PFWACXX:6:1101:1462:1956
@ERR2184190.sra.2 SBS123:200:C3PFWACXX:6:1101:1487:1970
==> read2.fq <==
@ERR2184190.sra.1 SBS123:200:C3PFWACXX:6:1101:1462:1956
@ERR2184190.sra.2 SBS123:200:C3PFWACXX:6:1101:1487:1970
Read name in alignment file
You may notice that, both read names are separated by a white space. for general purpose, only the first part of read name (eg: ERR2184190.sra.1) are saved in alignment file (bam file).  
Here are example for your reads. the first line is read1, and the second line is read2. the read name are identical (1-column), and they are separated by the FLAG field, (2-column).
# subset 100 reads from the file
$ bwa mem Oaureus.fa read1.fq read2.fq | samtools view -Sub -f 2 - | samtools sort -o aln.bam - 
$ samtools view aln.bam | head -n 2
ERR2184190.sra.37   99      VASH01007726.1  2202973 60      101M    =       2203373 501 ...
ERR2184190.sra.37   147     VASH01007726.1  2203373 60      101M    =       2202973 -501 ...
                    
                
                 
EBI-ENA has the fastq files clearly marked as R1 and R2. You can get the data from there.
to my understand, the latest illumina output fastq file contains
two partsin the name line, separated by a "white space",the name+index information. the name part are identical for read1 and read2. Only the first part will be used in further analysis, that refer to this read.The following are what I get from HiSeq XTen platform.
Last, your "first 2 headers" are not like the real output:
The name of reads in read1 and read2 must in the same order, or else, the aligner will report errors and stop working.