Depends on the type of bridge. An older technology called WDS (Wireless Distribution System) was never a wifi certified protocol, so implementations vary, leading to incompatibilities, and so often don't work across brands. That’s probably what those ppl were referring to.
In more recent years, the industry has moved to "universal" bridging. In this case, the bridge connects to the remote AP as a client using wifi standard protocols (B/G/N), so it’s just as compatible with your router as any PCI/USB wireless adapter using those same standards. If the bridge also supports repeating (some do, some don’t), then it can establish its own AP for that purpose.
The link I provided is a universal wireless bridge (it was mere coincidence I provided Netgear as an example). It only supports a wired client, it does not support repeating.