For my big games, I usually will have after a short time three kinds of ships (sometimes four): All-purpose Attack/Defense ship, All-purpose Transport ship, Carrier, and (sometimes) Point-Defense ship. I'll just use normal SEIV naming conventions (ie Ship XII 0042, meaning it's the 12th revision on the basic design, and the 42nd ship constructed on that design).
For more "intimate" games, like SJ plays, I name my ships. My most recent one was playing with his P&N mod, I had at the end, 5 BBs (Victory, Privateer, Protector, Bucaneer, Thief), 3 DSs (Skull, Crossbones, Eyes), 2 LCs (Eliminator, Vengance), 1 CR (Refueler), 1 DN (Miner), 4 SS (Shredder 1 through 4), 1 BS (Shredder 5), and I at various points used captured enemy DNs in battle before analyzing/scrapping them, they were temporarily named "Bounty X", with X being whichever number it was in order. There were fifteen, I believe. The BBs were all of the same general design, heavy shielding, PPB Vs, a few PDLasers. Victory had a small SY, and reached 50% experience, Privateer reached 35%. Skull and Crossbones are twins, moderate shielding, HBM V, SD V, PPB V, BP V. Eyes had SD V, 3x PPB V, and HyperOptics III. Eliminator was a better shielded Skull/Crossbones. Vengance was a heavily shielded delivery vehicle for a Plague Bomb V. Refueler was all engines, supply storage, and solar collectors. Miner was a captured DN retrofitted to do mining work on asteroids. Shredders 1 thru 5 were mining bases.
For you to think, "I'm sending Victory into battle against three enemy DNs," it is so much more fun than "I'm sending BLaster XV 0174 into battle against three enemy DNs." But it's really impossible to do that for larger games except for special cases (ie, ships that have reached Legendary). Makes for good story writing in small games too