Did the Portuguese and Japanese recognize that they both played chess after contact?

Probably.

First off, the rules of shogi and chess have changed hardly at all since the Portuguese landed in Japan in 1543. Hence I think I can make my case purely by explaining the similarities and differences of the two games in their modern forms.

Shogi looks like this

Chess looks like this

Strategic Similarities

The two games are built and play very similarly on account of four attributes. However, in my experience, virtually any game with these properties will "feel like" chess in that roughly the same strategies can be used (i.e. forks, trading up, sacrifices, planning ahead, etc.).

The objective of the game is to use your army to capture the opponent's leader.
Non-king pieces have no intrinsic value except to attack and defend the leaders
Turn based
Perfect information

Piece movement and setup

That's not really what would tip you off though. What would in my opinion is how the pieces move.

Pawn / 歩兵 fuhyō

Spawns in the front row
More numerous than any other piece (8 vs 9)
Standard moves up 1
Pawn is derived from the medieval latin peon for foot soldier; 歩兵 fuhyō literally means foot soldier

Rook / 飛車 hisha

Moves n up/down/left/right

Bishop / 角行 kakugyō

moves n diagonally

Knight / 桂馬 keima

keima jumps two up + one left/right (but not sideways or backwards like a knight)
Starts in the second position in the back row
Japanese name literally means horse

King / 王 ōshō (玉將 gyokushō for the lower ranked player because social heirarchy needs reinforcement even in board games)

Moves one up/down/left/right/diagonal
Starts close to center/center in the back row

Pieces that do not have good equivalents

Queen
銀將 ginshō
金將 kinshō
香車 kyōsha

Small details

Both games allow for a promotion when certain pieces reach sufficiently far into opposing territory, but this is likely a pragmatic mechanism to counter pawns and shogi pieces problems moving backwards.
Shogi has a "drop rule" where you can spend your turn placing a captured piece under your control. This only exists in certain relatively modern chess varients like bughouse chess
Shogi is played on a 9x9 board as opposed to an 8x8 board

Conclusion

5/6 Chess pieces have Shogi equivalents while 5/8 Shogi pieces do. The setup is uncannily similar and the objectives are identical. Even the names of pieces maintain descriptive similarity in some cases. Given that chess was probably the most popular game in 16th century Europe, it would be surprising not to recognize the similarities.

Anecdote

I also play Chinese chess (xiangqi) reasonably well. I once taught a expert (elo 2000-2200) Chess player. Got stomped every game. Most chess variants are similar enough that skill transfers pretty quickly.