I usually freeze mozzarella and then grate it (longer the better, unless you are in a hurry, then 20 minutes or so works OK). This works very well. Other soft cheeses, such as those meant to be eaten at room temp, brie, for example, I wouldn't freeze... Of course, I don't think many of us are grating brie anyway.
The most similar, of course, the one you could easily find in any supermarket, is the "Camoscio d'Oro". This is not similar but the same, since is made by Bongrain SA, a French food group.
As semi-industrial cheese made in Italy, we have the "Alpino", better or similar to "paglietta", both of Osella, which is sited in Piemonte (Turin), region on the border with France.
But all over the region there are many small firms dairy, which produce the "tomini". Tomini are a tipical regional cheese, smaller but very similar to brie, both as flavor, as texture and as aging. some news
You can find tomini freshly made, very sweet and tender, and, day by day, more and more seasoned tomini. The last ones can be done on the grill.
If you switch to production typical, you can find a variety of high-quality productions of tomini and similar products in many farms in the region. Like "Montebore" or "Bra", or very similar products that, if you are lucky, you can find among the reserved ones from some kind local farmer.
We share a border, mountains, traditions, people. We even have a whole region half Italian and half French, the Val d'Aosta. Incidentally also some cheese!
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Dry ricotta or even dry cottage cheese are similar. If you have access to an Indian grocery, paneer is a somewhat comparable choice.