Are unique node titles neessary? I am considering automatically assigning the uid to several of my content types automatically.  However, one other data type uses a number as its title too (product number actually.)  There is a strong potential for a uid to be the same as a product number.  Could this potentially cause problems?  What are some common problems that can arise from non-unique titles?
Drupal itself does not depend on titles or labels to identify entities, and there is no expectation within the core system that they would be unique. You could in theory have a completely functional site with ten thousand nodes all sharing the same title, tagged with ten thousand terms sharing the same name, linked by ten thousand menu items sharing the same label.

Drupal is full of locally unique, automatically generated IDs: each node has an nid, each term has a tid, each menu has an mlid, and so on (though for identifiers unique across the system, one must use UUID or something similar). Provided your custom code, theme output, Views filters, and so on can be written to reference these machine IDs, I would expect few problems. 

It is not always possible to rely on those machine IDs; they are content, and so are subject to editing, deletion, and replacement. You might want a banner to appear on your About Us page, but depending on the season the About Us page may be node/25 or node/29, for example.

Thus it is possible, even unconsciously, to build parts of the site that do depend on labels. Many themes insert a class on the &lt;body&gt; based on title, or on the alias/context (which pathauto will by default set to the title); you may have CSS rules applying to the wrong pages. You may have an editorial policy that all nodes pertaining to a particular product include its product code in the title, and set up Views to filter them. But only takes one typo for a node to go missing, or one forgotten edit to the title of a cloned node to have extras appearing.