Your adopted brother has tested with both AncestryDNA and with 23andMe and has transferred his raw data to FTDNA. The ISOGG Wiki can tell us which tests each company provides and which tests which can be transferred.
In the AncestryDNA article, under Transfers, the Wiki says:
AncestryDNA results can be uploaded free-of-charge to the third party
website GedMatch where many additional tools are available for
advanced analysis. For more information on GedMatch and other
third-party tools see the Wiki article on autosomal DNA tools.
AncestryDNA results can also be transferred to Family Tree DNA's
Family Finder database where you can search for additional matches,
join projects and use additional tools to compare results. The
transfer is free but a small fee is required to unlock additional
features. Details can be found at
www.familytreedna.com/AutosomalTransfer. Note that AncestryDNA
transfers after the introduction of the new chip in May 2016 will
receive a reduced number of matches and will not receive matches in
the speculative range. See the blog post from Louise Coakley Should
you upgrade your Family Finder transfer?.
For 23andMe's test history, the wiki says:
A range of different chips have been used for the 23andMe test since
the introduction of the service:
v1: November 2007
v2: September 2008, ~555K SNPs
v3: November 2010, >900K SNPs
v4: November 2013, ~570K SNPs
v5 August 2017, ~640K SNPs (Illumina Global Screening Array)
Tests done using the version 3 and version 4 chips can be transferred
free of charge to the Family Tree DNA Family Finder database. For
details see the FTDNA page on the autosomal DNA transfer program.
While you can get some Y-DNA data from an AncestryDNA transfer (see the section Extracting Y-DNA data in the AncestryDNA article), including a haplogroup assignment, you don't get the same amount of data as someone who takes a full-fledged Y-DNA test like the ones that FamilyTreeDNA offers. (See their article Y-DNA Basics.)
Could it be that your brother's half-brother took a Y-DNA test at FTDNA? If that's the case, they wouldn't match because the half-brother's results are not in the FamilyFinder Database (which has the results from autosomal DNA tests).
If my guess is right, and:
- They are in two different databases on FTDNA's site, and
- FTDNA doesn't match the results of Y-DNA tests to the autosomal database for you
-- then the only way they could match at FTDNA would be if they both participated in the same surname project. Joining projects is something you do voluntarily -- FTDNA doesn't submit your data to those projects without asking you.
So this is my long-winded way of saying that without knowing exactly which tests both of them took, we can't answer your question with any certainty. The previous answer assumes that both took autosomal DNA tests, but that might not be the case.