Sounds like you want to at least start with de-duplication and data standardization. There is a huge variety of stuff out there both commercial and opensource. My personal tool of choice is Talend (They have a tool for almost everything related to data and an active community) you can download their opensource profiler - Talend Studio for Data Quality here: http://www.talend.com/products/data-quality. You can use this to profile the data in all sorts of interesting ways and then develop jobs to de-dupe and standardize your data. Sounds like you have a long road ahead of you, have fun!
Talend Data Integration Tool. It's free and supports almost all DB's going both ways. I've been using it for over a year. It's SOOOO much faster than all the other tools I've tried, including the paid ones. It's extremely flexible but I suppose with that flexibility comes a little price in manual set up where a lot of other tools will automatically map things for you. Check it out and if you have any other questions I'd be happy to help: http://www.talend.com/products/open-studio-di.php
While this is surely overkill, Talend Open Studio can do this. It is a free ETL solution. It's definitely a sledgehammer for this task, Foreign Data Wrappers in PostgreSQL are definitely better in terms of performance. But being someone who is newish do DBs, having a PointyClicky GUI interface might help.
As an ETL, you can define data sources (MySQL in your case), transformations (which might come in handy for data type conversions) and data "sinks" (PostgreSQL in your case). You then just connect the visual components together and hit the big green Play button.
Edit: They don't have many screen-shots on their home-page, but a Google images search should give you a good idea.
I have used Data Wrangler before (http://vis.stanford.edu/wrangler/app/) and OpenRefine. But typically, if the dataset is not too large, I tend to use Weka. There are many ETL tools our there too (e.g., http://www.talend.com/resource/etl-tool.html) but I've not used them before. Anybody have any good open source ETL tools to recommend?
The tool looks at first sight very interesting, but also very tied to bioinformatics. I wonder how easy it would be to adapt Pathomx to different domains.
In the area of ETL I'm looking at the moment for an alternative to the heavyweights Talend Open Studio for Data Integration and Pentaho Data Integration. Would Pathomx be a possibility?
As I have seen the creation of plug-ins is easy, but I could also hide the existing bioinformatics tools?
Magento has the ability to do this but like other's have said, it's not a toy and depends on how serious you are and what level of integration you need.
There's a sync for MYOB and Quickbooks and we're about to undertake some development to integrate with Navision (sorry, MS Dynamics Nav).
There's a product called Talend that can synch one way or two data between disparate systems but Talend is it's own beast. Magento also has some pretty decent add-on extensions that can spit out your sales /inventory data in daily batches to different formats (csv, xml etc) and ftp it or place it in a directory for an ERP / Accounting system to import.
Not web-based per se, but check out Talend - http://www.talend.com/. The open source version is highly capable and can grab files from FTP, HTTP, email, disk, etc and load them to just about any datastore out there.