Convert concrete stairs to limestone stairs
I believe it can widened if you connect the concrete pieces together (drill holes and add rebar between the two). Still, you may get a crack that would probably carry through any tile/grout you put on it.

I'd instead consider replacement of the concrete with a wider piece. Either prefabbed or poured on site. If you pour it on site, you could have the concrete stamped and stained to have that nicer look.

You could also consider doing wood over the existing and widen it as wanted. You will want to pay attention to the step height and the supports for the edges of the wood. If you don't want the maintenance, composite decking would work for that.

I'd also consider a railing regardless. Tile if used can become slippery when wet and if you live anywhere snow/ice can occur it could become an unpleasant ride down.