It is actually pretty much fluff in BT that everyone fields everything, with few exceptions such as factional variants or rare factional exclusives such as Dragons for Draconis Combine or Ravens for CapCon, but thanks to salvage even those can be common around certain parts (such as Dragons not being perhaps uncommon among FedSuns units that garrison the Draconis Combine border).
At least around 3025. Post Clan Invasion there are far more faction-specific 'Mechs.
The mixture of 'Mechs is the legacy of the Star League armies, and the Succession Wars. The Star League used the best 'Mechs, and equal or downgraded variants of these were used by member states, so everyone had similar 'Mechs overall. And once the Succession Wars began, 'Mechs were even more mixed up due to salvage, captured examples, captured factories, etc.
If you want clearer faction identity when it comes to 'Mechs, i recommend looking at later eras, for example by the Civil War (3062-3067) most factions have good selection of mostly-exclusive designs.