You can certainly paint tiles rather than replacing them. The Dulux Renovation Range is a very popular choice for painting tiles. Feedback on the product’s longevity has been fantastic if it is applied as per the manufacturer's specifications.
Wash the tiled surfaces at least twice with sugar soap and rinse well.
If you follow the directions above you'll end up with a paint that is strongly adhered to the underlying tile and sufficiently protected from scratching and abrasion. - MitchellMc
A few years ago I used White Knight brand. It came up nicely. There is a White Knight primer as well, which you need to use. Importantly, ensure that the surface is very thoroughly cleaned before paint application.
At that same time, I painted the inside surface of the bath too (it was a shower-bath). That didn't go as well, because we were standing on the painted surface to have a shower and it eventually peeled off (after a few months). So eventually I scraped back the paint off the bath, but left the wall tiles painted. - jaga
I did it myself on kitchen tiles. Prep is the key here. Spend a heap of time thoroughly cleaning, cleaning, cleaning and then sand too. But note with sanding you're only sanding to help paint adhere to existing tiles, so no need to be as aggressive with sanding ...cleaning is a different matter and is the key here. - KevinMall