No. Adobe only builds Bridge for Windows and Mac, there is no browser version, and nothing Adobe offers will install it on Linux. digiKam does the same filing and sorting job for free.
Adobe Bridge by Adobe · Photos & graphics · No way to run it
Adobe Bridge is a Creative Cloud app and Adobe's technical requirements list Windows and macOS only.
There is no Linux installer and no web version, so unlike Photoshop there is not even a browser
fallback.
Bridge was never really an editor anyway. It is a filing cabinet: you browse folders of photos, rate
and label them, add keywords, rename in batches and hand the picked ones off to Photoshop or Camera
Raw. That job has a good free replacement on Linux.
digiKam is the closest match. It reads your folders, shows thumbnails, handles star ratings, colour
labels, keywords, faces and batch renaming, and it writes that information into the standard photo
metadata rather than a private database. If you preferred a lighter browser with no library at all,
XnView MP behaves more like the old Bridge window.
Your ratings and keywords usually survive the move. Bridge writes them to XMP sidecar files or into
the photo itself, and digiKam reads both. Collections and saved searches that only lived inside
Bridge do not come across, so note those down before you wipe Windows.
Free photo organiser with ratings, colour labels, keywords, face tagging and batch rename. More of a library than Bridge, so expect a little setup.
Your files: Reads star ratings and keywords from XMP sidecars and embedded metadata written by Bridge. Bridge collections and saved searches do not transfer.
Worth it only if you used Bridge as a doorway to Camera Raw, because darktable combines the browsing and the raw editing in one place.
Your files: Reads raw files and XMP sidecars. Camera Raw edits themselves do not carry over.
This doesn't have to stop you
One stubborn program doesn't have to decide the whole question. You can keep Windows on the same PC and restart into it only when you need Adobe Bridge. Everything else you do can move.