Can a Buddhist mix Buddhism with other religions?

Buddhism is not a religion. What Buddha revealed to the world is, how the world exists, how the things happens. And also interestingly, since the world is not a place which gives you ultimate happiness, Buddha has shows us the way to uproot the existence of this life.

So if you want to have other religion and buddhism there's no issue. But when you realise the core of the buddhism, you start to move away from all the religions as those are useless to the ultimate happiness or the Nirvana

Whether Buddhism is called a religion or not, the thing to consider is do the two systems of thought contradict each other? If there is contradiction then one cannot follow at least one of the religions. For example, Buddhism and Christianity contradict each other so you could not be a Buddhist-Christian. But you may be able to be a Buddhist-Quaker (in the liberal since) depending on what you took from Quakerism, since they have no creed. There are plenty of examples of how Buddhism has absorbed ideas from different countries. However, there are no texts which grant or deny the possibility of having multiple religions. I think it is just not probable since Buddhism is so much different from what we normally call religions.

Personal conclusion::

Dhamma gives me wings, free me from blind-faith, free me from sensual-pleasing-tradition-bounds, ... enable me to be more powerful.

Religion whether it be Hinduism or buddhism or sikhism or christianism or jainism or muslim etc. binds me to blind-faith, binds me to sensual-pleasing-tradition-bounds,... makes me weak.

To quest for authoritative sources

All the suttas were composed 100s of years after buddha. So how much originality is still left, can't be claimed.

Still, you may see this pre-sect-buddhism which is considered to be original teaching, still it is also considered to be mix of vedic, jain and sramanic schools of thoughts.

Along with above, you might be interested in Mahayana, which in itself is a mixture of various thinkings, rituals, traditions, worships etc.

To get to know how people, who call themselves to be on the path of dhamma, are engaged in various practices which can be easily found in other religions also which were either developed far from dhamma-touch or before buddha or after buddha can be seen here too.

So now it's up to you whether you wish to remain within traps of religions or want to live freely under dhamma's guidance...

Still, it's the practice and right understanding through which you will be able to claim dhamma.

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To give a kick-start, you may go for Majjhima Nikaya.

My ego-satisfying-view claims that Mulpariyaya, sabbasava, dhammadayada's 2nd half, bhaybhareva, Vattha, sallekha, sammaditthi, satipatthana, are as per dhamma.

It is because my ego thinks that, "1st half of dhammadayada considers self-respect-issue(1 of the 8 worldly concerns), anangana lacks compassion, aankhankheya depicts somewhat maintain/gain-respect-issue. These might encourage some to practice more dedicatively but it will start with either greed or fear and work started for worldly concern is not a Noble path.

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Buddhism doesn't accept false views such as those listed in DN 1.

Many religions with soul and God reinforce the self view, the view of eternalism or other false views.

As such, it is not compatible to mix other religions with Buddhism, if they have false views according to DN 1.

It seems significant that HHDL advises us not to abandon the religion of our birth yet also encourages us to strive for realisation. This is the view that religion is a vehicle, not a destination.

All those I know who endorse more than one religion also say, equivalently, that they have no religion. It depends whether we see our religion as a proscribed set of beliefs or a set of methods and practices for transcending belief.

And then, talk to any three Christians and they will probably have three different religions, such is the game of Chinese Whispers.