EU OPS allows to have two active typeratings. However if the types are as different as boeing and airbus are you need to do all checks in each one, which means 8 training/check events per year, 2 line checks. Additionally one cannot fly both types in one duty period (working day).
And then there are some special cases. One is a "common typerating", which is the way boeing makes it easier to fly different types. 757/767 is one example, the other one is 777 and 787. The typerating entry lists both types, however one need a conversion training to be allowed to fly both.
The other one is Mixed Fleet Flying (MFF), which can be both used on boeing types and airbus, however is more likely to be experienced on airbus types. For example A320 family and A330, or A330/A340, A340/A380 although the latter one isn't used yet as far as i know. On the boeing side it isn't really used, however there is a (european) approval for 737/757 and 737/767 MFF. Boeing works on 737/787 MFF, but that isn't approved yet. The training from the first to the second type is called Cross Crew Qualification (CCQ) and is usually an abbreviated training as a full type rating is not needed.
On business jet types it is quite normal to fly several types, however one usually needs a full type rating for each.