Could sidesticks be linked?
Yes. Gulfstream G500 and G600 are the first civilian transport category aircraft to have active sidestick. However, instead of mechanical linkages between the two sticks, the side sticks are coupled electrically. It's much more prevalent in military aircraft.

Source: Aviation Week

So why isn't it being adopted by aircraft manufacturers with side stick controls (e.g. Airbus, Bombardier)? It pretty much all has to do with costs.


Active side sticks introduce more failure modes, such as jam and runaway, which would involve more software and mechanisms to detect and mitigate. These add complexity and introduce failure modes in their own rights, and they all add up to costs.
There is very little certification mandate to make side sticks active. All the objectives, such as stick force/G, stick force/V, have been met by passive side sticks. Airbus introduced envelope protection, most likely not because it was a nice-to-have, but because certifying the combination of lack of aerodynamic feedback on the inceptors and the choice of a pure C* control laws without speed stability demands it.
Certifying the first civilian active side stick is costly (but now it's been done by Gulfstream). There needs to be clear market reasons to do so. For Gulfstream, it may be (personal hypothesis) that the aircraft is being catered to the ultra-rich clientale and a novelty factor. For commercial aviation, the only reason I can think of is safety, which brings us to...
Would active side stick actually increase safety by leaps and bounds? This is a topic under research so the discussion has personal bias. On the A320 and A220, if both PF and CP side stick commands are detected, an aural "Dual Input" warning would sound. Obviously, AF447 crew ignored it. The final BEA accident report didn't mention the lack of tactile feedback as a reason for the crash; rather, crew confusion on stall recovery and lack of airspeed disagreement warning were highlighted.