Can a synthetic brush cause irritation?

[luvmysuper]

New allergies can be unwelcome surprise | CNN

Skin reactions to fragrances and perfumes are common, but to have a reaction from finished plastic products like plastic bags is extremely rare.

edition.cnn.com

Not so much the plastic, but what's IN the plastic.
If plastic is reused without clearing all non volatile compounds, say a container that held Dimethyl Fumarate (DMF, an antifungal used to prevent mold during shipment commonly used in China) then skin issues can result.
In any case, pretty easy to isolate the cause by trying the same cream with a different brush.

 

[BigJ]

Try the same cream with brush you have that didn't give you spots, and try the same brush with a cream you have that didn't give you spots. That will give you the answer.
I wouldn't expect a lot out of a 5 dollar brush though.
Depending upon its country of manufacture, you may have no idea what those bristles are actually made of.

+1! Great advice! ‘I wouldn’t expect much from a $5 brush..’ At worst might work as a travel brush.