Is the word monte the same in Brazilian and European Portuguese?

No.

While the literal meaning of "monte" (a pile, a heap, or a small hill) is identical in both dialects, there is a difference in typical usage. In Brazil, the phrase "um monte" is extremely common as a colloquial way to quantify things, functioning as a synonym for "a lot" or "many." While a Portuguese person understands this usage, they are much more likely to use "muito," "bastante," or "imenso" to express the same idea.

Brazilian Portuguese usage

1. Eu tenho um monte de trabalho para fazer. (I have a lot of work to do.)
2. Ele fala um monte de bobagem. (He talks a lot of nonsense.) /3. Tem um monte de gente na festa. (There are a lot of people at the party.)
3. Eu comprei um monte de doces no mercado. (I bought a lot of sweets at the market.)
4. Eu gosto um monte de viajar. (I like traveling a lot.)

Continental Portuguese usage

1. Eu tenho muito trabalho para fazer. (I have a lot of work to do.)
2. Ele diz muitas asneiras. (He says a lot of nonsense.)
3. Há muita gente na festa. (There are a lot of people at the party.)
4. Comprei imensos doces no supermercado. (I bought loads of sweets at the supermarket.)
5. Gosto muito de viajar. (I like traveling a lot.)