Does AWS Aurora suffer from the same limitations that MySQL suffer from?
There is no limitation of 64KB on the size of a MySQL row.  There are limitations, but almost never does anyone hit them.  Can you point to where you got that limit?

There are a couple of obscure limits in InnoDB, such as having dozens of big VARCHARs, but they are easily worked around.

Here are the main limits for MySQL:  http://mysql.rjweb.org/doc.php/limits

Edit

At first glance the manual page on row size says there is a 64K limit on row size, but reading further it says "BLOB and TEXT columns count from one to four plus eight bytes each toward the row-size limit because their contents are stored separately from the rest of the row."  Since LONGBLOB and LONGTEXT columns can contain up to 4GB each, the maximum total row size (including such columns) is at least 4GB.

Furthermore, "stored separately" is dependent on the ENGINE involved; that quote does not cover all the cases.

The bottom line is that it is possible to create a table with more than 64KB of data.  (I don't know about exceeding 4GB.)  Doing so may require using TEXT instead of VARCHAR(65535) (which is virtually identical in semantics), etc.

I have made many tables (MyISAM and InnoDB) where the "row size" was bigger than 64KB, both in definition and in fact.  I have multiple cases of MEDIUMTEXT or MEDIUMBLOB that work fine, back at least as far as version 4.0.