Matching Original Tile, and the Honest Alternatives
Close Is Worse Than Different
Tile from the middle of the last century was made in sizes, thicknesses and glaze colours that are not produced now. A near match reads as a repair from across the room, because the eye detects a small mismatch far more readily than a large one. Anything that is not an exact match should be an obvious choice instead.
Salvage
Architectural salvage yards and online sellers do carry period tile, and patience is the main requirement. Where a wall has to be opened for plumbing, taking tile from a low or hidden area to patch a visible one, then using new tile in the hidden spot, is the oldest trick available and still the best.
A Deliberate Band
Rather than patching a hole, remove tile to a clean horizontal line and introduce a contrasting band or a different material above or below it. The result looks designed rather than repaired, and it converts an unsolvable matching problem into a straightforward layout decision.
Refinishing Instead of Replacing
Where the problem is colour rather than damage, professional refinishing of tile and of a cast iron tub is a real option. It is a coating with a finite life measured in years rather than decades, and it needs a specific cleaning regime afterwards, but it keeps the original substrate and it is a fraction of the cost of a strip out.
Grout Alone Can Do a Lot
Sound tile with tired grout looks far worse than it is. Raking out and replacing grout, and replacing every silicone joint at the same time, transforms the appearance of an old bathroom for the price of a day of labor and a few materials. It is worth doing first, before deciding what else the room needs.