NO. Rather, the tag description for oracle should be updated to be more specific, like the StackOverflow description.
In the context of this site (and in the general vernacular of developers/DBAs), I would expect oracle to refer to the product Oracle Database, rather than the comapny. I imagine this is how most people view the word "Oracle" in a database context, and widening the definition to include MySQL can start causing confusion for anyone asking/answering if they haven't read the tag description...And if a person thinks they already know what "Oracle" refers to, they probably won't read the tag description. How many of us have bothered to read the tag description for select? Ignorance is not an excuse for mistagging a question, but tags should still be intuitively understandable.
In terms of precedence for this type of tag, I should point out that there is no microsoft tag. If we allow both Oracle Database and MySQL questions to be encompassed under "Oracle", then we ought to allow SQL Server and Access to be encompassed under "Microsoft". On the other hand, there is a sybase tag, even though they have multiple products such as ASE, IQ, etc. However, the sybase tag description does explicitly say to include a tag of for the actual product.
Would it add any value to a question if the oracle tag referred to all of the company's products, rather than Oracle Database? If so, would that value outweigh the cost of either re-tagging old oracle questions to include oracle-database (like the sybase tag), or conversely, simply leaving old questions with the now-ambiguous tag?
Lastly, we should keep in mind that DBA.SE shares a lot of users with StackOverflow, where it's very well understood that the "Oracle" tag refers to Oracle Database. I'm not saying DBA.SE has to mirror SO, but consistency across the Stack Exchange network is definitely a positive.
