max number of content types on drupal
560 tables isn't that much (D7 produces a lot more because of one field => 2 tables) but 150 content types is a lot, that's for sure. This will result in huge caches that Drupal needs to carry around all the time.

I assume that site uses content types to store any kind of data, including internal stuff that isn't displayed?

Using content types for anything that is displayed on the website usually makes sense, you also get a lot of integration for free (e.g. views). However, having a huge amount of content types, doesn't scale well as you found out so you might want to go through your content types and convert some of them to custom tables.

In Drupal 7, it is common to define entities for custom data structures, especially things that are not content (see drupal commerce, search api and so on for examples).

There, is, however, no official or technical limit of content types that you can have. I'd recommend using xhprof to profile the site, if you can e.g. show that the site is using a huge amount of memory because of that, that might be a good argument.