Regarding OnlyOffice‘s legal claims, this is the response by the FSF, which is the copyright holder to the GPL licenses . It is their opinion, that these specific license additions by OnlyOffice are not legal.

[…] In the main repository of the OnlyOffice DocumentServer, we have found that the README file (and similar README files located in other OnlyOffice repositories) clearly state that the software is made available under the AGPLv3 in the "License" section. However, OnlyOffice then includes additional terms in the LICENSE file (and in some other LICENSE files in other repositories), as well as in license notices of individual source files. In utils.js, for example, it states: "Pursuant to Section 7(b) of the License you must retain the original Product logo when distributing the program." This obligation to "retain the original Product logo" is not included in Sec. 7(b) of the (A)GPLv3, nor in any other parts, as an (A)GPL-compliant additional term, and is therefore considered a further restriction of the (A)GPLv3. 

The (A)GPLv3 makes it clear that it permits all licensees to remove any additional terms that are "further restrictions" under the (A)GPLv3. It states, "[i]f the Program as you received it, or any part of it, contains a notice stating that it is governed by this License along with a term that is a further restriction, you may remove that term." […]