It is general wisdom that a key should only be used for a single purpose. Using the same key for multiple purposes is like swimming in shark-infested waters with a leg wound. It's not guaranteed to kill you, but that doesn't make it a good idea.
One of the things that can go wrong is if the same construction has a different meaning in the two systems. For example, in system A, signing the message $M$ means “I give Bob 42 units”, and in system B signing the same message $M$ means “I give Charlie 24 units”. Many cryptosystems use uniquifiers to avoid this kind of problem, for example by only ever accepting messages that start with "in system A:". If both systems have uniquifiers that ensure that no message can be valid for both systems, then using the same key is ok with respect to this pitfall. I have no idea whether this is the case for Matrix and Solana.
A more subtle indirect variant of this pitfall is consequences of implementation flaws. For example, maybe the software for system A has a side channel vulnerability, but it's only a partial leak that doesn't provide enough information to construct a useful attack. Meanwhile system B runs on a platform that's weak to glitch attacks, but there's no practical way to trigger a glitch that's useful for system B because it's been constructed to defend against those glitches. However, the partial information from system A is enough to construct a useful glitch that breaks system B. This is basically a more subtle, but more realistic of “you can extract the first half of the key from system A and the second half of the key from system B”. A lot of real-world attacks against systems that people have seriously tried to secure are like that: exploiting a combination of flaws, each of which does not result in a meaningful attack on its own. Sharing secret material between systems increases the risk.
Another reason to avoid using the same key for different purposes is that if the key is compromised, this increases the impact. Somebody manages to distribute a fake Matrix client that extracts private keys, and oops, now the attacker also has your Solana key.
Besides the benefits you list are actually quite dubious.
reduction in key management
In practice, probably not. You'll have to feed the key to both the Solana software and the Matrix software, meaning more exposure for the key than if each key is only known to one of the systems. Most keys are kept online and usable, not in secure vaults where they are only accessed once a year with ample prior notice.
being able to re-use identities tied to existing public keys.
Tying identities to public keys is in itself a bad idea. It makes it very painful to revoke the key: you have to change identities! There's a reason PKI exists. Tying identities to public keys also negates privacy.
