Products with 2 different titles
Yes, a product can have two different titles.

You can add fields to products on the product type admin screen (/admin/commerce/config/product-types) by choosing the appropriate product type and selecting manage fields.  I would add a text (plain) field and call it short name or something like that.

Then, you can configure the display settings for the product type to hide the short name field (disable it), ensuring only the full title is shown on the product page.

Then, in your grid product catalog, configure your view (or whatever you use to make the catalog) to show only the short name, not the title field.