Almost every coin has its own infrastructure, so yes, you will need another wallet for almost every altcoin.
Luckily, there have been some people who invented hardware wallets which simplify the process.
Meaning you can hold multiple altcoins in this one program.
They still have their own wallet, but is just much more user friendly.
These wallets are both open source, and you are the only one who has acces to your private key, which means that these are pretty secure.
Supported Coins (64!!): BTC, ABN, AC, AUR, BTA, BELA, XBC, BLK, BRIT, CDN, CCN, CLAM, CLUB, COMP, DASH, DGB, DOGE, DGC, EDR, EFL, EMC2, ETH, ETC, EXP, FTC, GAME, GCR, GRC, NLG, INSN, IOP, JUM, LDCN, LBRY, LTC, MONA, XMY, NMC, NAV, NEOS, NVC, USNBT, NSR, OK, PKB, PPC, PSB, PIVX, POSW, POT, RDD, RICHX, RBY, SDC, SMLY, SLR, STRAT, SYS, UNO, XVG, VRT, VASH, ZEC, XZC
Supported coins (12): BTC, LTC, ETH, DASH (not on iOS), ETC, REP, ZEC (not on iOS), DOGE, ICN, GNT, GNO, DGD