13 feet is a long but acceptable span for 2x8s at 24", with the thinner 5/4 deck boards. The (apparent) lack of ledger might be unusual, but if joists are properly spaced and properly supported by the brick wall, that's probably a non-issue.
If you're experiencing bounce, two ways to reinforce are to add blocking between joists midway between the house and the beam, or using 2x deck boards instead of 5/4. Thicker decking has a not-insignificant additional weight, which should still be OK with your 2x8 joists at 24" spacing, but reinforcing the joists as well is probably necessary.
However, it also sounds like you're considering replacing the decking with composite; your 24" joist spacing is likely insufficient for typical composite which requires tighter joist spans usually (mine is at 12", but 16" might be allowable). You'll get more mileage out of composite if you also replace beams and joists, especially if they're in bad condition (the photos aren't great, but it looks like your joists are old). Composite deteriorates very slowly, but wood (cedar, pressure treated) much more quickly so composite deck failures are usually on the wood supports.
If you replace the entire deck, you can have the plan for the new one evaluated for proper joist depth (based on the span given in an edit it likely should be 2x10 not 2x8) and spacing (dictated by the type of decking surface you use) as well as address any concerns you have with the house connection, putting the new one directly on brick like this one or using a proper ledger.